[Nexus 7 (2013) Android Development] Alucai Vivorvel:
Alright, so this time, after the orientation got stuck again, I recorded me doing a combination of rotating the device, turning off the screen, and locking Auto Rotate (in different orders) via CatLog. Filtering for Errors.

05-28 10:17:00.047 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:00.047 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:00.217 E/memtrack(29956): Couldn't load memtrack module (No such file or directory)
05-28 10:17:00.217 E/android.os.Debug(29956): failed to load memtrack module: -2
05-28 10:17:00.287 E/memtrack(29969): Couldn't load memtrack module (No such file or directory)
05-28 10:17:00.287 E/android.os.Debug(29969): failed to load memtrack module: -2
05-28 10:17:00.578 E/memtrack(29977): Couldn't load memtrack module (No such file or directory)
05-28 10:17:00.578 E/android.os.Debug(29977): failed to load memtrack module: -2
05-28 10:17:00.618 E/ACDB-LOADER( 219): Error: ACDB AudProc vol returned = -8
05-28 10:17:00.768 E/memtrack(30005): Couldn't load memtrack module (No such file or directory)
05-28 10:17:00.768 E/android.os.Debug(30005): failed to load memtrack module: -2
05-28 10:17:01.268 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:01.268 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:02.490 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:02.490 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:03.721 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:03.721 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:04.942 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:04.942 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:06.173 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:06.173 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:07.384 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:07.384 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:08.616 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:08.616 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:09.837 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:09.837 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:11.048 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:11.048 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:12.279 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:12.279 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:13.500 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:13.500 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:14.722 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:14.722 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:15.092 E/qcom_sensors_hal( 615): hal_wait_for_response: pthread_cond_timedwait() rc=110
05-28 10:17:15.092 E/qcom_sensors_hal( 615): hal_smgr_report_delete: ERROR: no response from request
05-28 10:17:15.652 E/ACDB-LOADER( 219): Error: ACDB AudProc vol returned = -8
05-28 10:17:15.943 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:15.943 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:17.174 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:17.174 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:18.395 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:18.395 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:19.406 E/qcom_sensors_hal( 615): hal_wait_for_response: pthread_cond_timedwait() rc=110
05-28 10:17:19.406 E/qcom_sensors_hal( 615): hal_smgr_report_delete: ERROR: no response from request
05-28 10:17:19.606 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:19.606 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:20.777 E/SensorManager( 823): sensor or listener is null
05-28 10:17:20.828 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:20.828 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:22.059 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:22.059 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:22.459 E/qcom_sensors_hal( 615): hal_wait_for_response: pthread_cond_timedwait() rc=110
05-28 10:17:22.469 E/qcom_sensors_hal( 615): hal_smgr_report_add: ERROR: No response from request 1
05-28 10:17:22.469 E/qcom_sensors_hal( 615): hal_acquire_resources, hal_smgr_report_add() failed, handle=0
05-28 10:17:23.280 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:23.280 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:23.951 E/qcom_sensors_hal( 615): hal_wait_for_response: pthread_cond_timedwait() rc=110
05-28 10:17:23.951 E/qcom_sensors_hal( 615): hal_smgr_report_delete: ERROR: no response from request
05-28 10:17:24.301 E/ACDB-LOADER( 219): Error: ACDB AudProc vol returned = -8
05-28 10:17:24.501 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:24.501 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:25.722 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:25.722 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:26.953 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:26.953 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:27.714 E/qcom_sensors_hal( 615): hal_wait_for_response: pthread_cond_timedwait() rc=110
05-28 10:17:27.714 E/qcom_sensors_hal( 615): hal_smgr_report_add: ERROR: No response from request 1
05-28 10:17:27.714 E/qcom_sensors_hal( 615): hal_acquire_resources, hal_smgr_report_add() failed, handle=0
05-28 10:17:28.175 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:28.175 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:28.635 E/ACDB-LOADER( 219): Error: ACDB AudProc vol returned = -8
05-28 10:17:29.396 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:29.396 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:30.607 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:30.607 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:31.838 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:31.838 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:33.059 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:33.059 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:33.420 E/qcom_sensors_hal( 615): hal_wait_for_response: pthread_cond_timedwait() rc=110
05-28 10:17:33.420 E/qcom_sensors_hal( 615): hal_smgr_report_delete: ERROR: no response from request
05-28 10:17:34.281 E/SensorManager( 823): sensor or listener is null
05-28 10:17:34.281 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:34.281 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:34.971 E/qcom_sensors_hal( 615): hal_wait_for_response: pthread_cond_timedwait() rc=110
05-28 10:17:34.971 E/qcom_sensors_hal( 615): hal_smgr_report_delete: ERROR: no response from request
05-28 10:17:35.502 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:35.502 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:36.733 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:36.733 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:37.954 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:37.954 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:38.885 E/ACDB-LOADER( 219): Error: ACDB AudProc vol returned = -8
05-28 10:17:39.175 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:39.175 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:40.397 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:40.397 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:41.618 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:41.618 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:42.849 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11
05-28 10:17:42.849 E/qcom_sensors_hal( 615): hal_process_report_ind: Bad item quality: 11

Let me know if you prefer an attachment, if so I'll upload it next time I'm at my computer.

Edit: ah, nuts. I restarted the device after posting that, and it got stuck again, this time in the opposite orientation.

I don't think it's purely hardware related, because usually a restart fixes it, and other ROMs I've tried aren't nearly this bad.