  Recent observations of the periodic Fast Radio Burst source 180916.J0158+65
(FRB 180916) find small linear polarization position angle swings during and
between bursts, with a burst activity window that becomes both narrower and
earlier at higher frequencies. Although the observed chromatic activity window
disfavors models of periodicity in FRB 180916 driven by the occultation of a
neutron star by the optically-thick wind from a stellar companion, the
connection to theories where periodicity arises from the motion of a bursting
magnetar remains unclear. In this paper, we show how altitude-dependent radio
emission from magnetar curvature radiation, with bursts emitted from regions
which are asymmetric with respect to the magnetic dipole axis, can lead to
burst activity windows and polarization consistent with the recent
observations. In particular, the fact that bursts arrive systematically earlier
at higher frequencies disfavors theories where the FRB periodicity arises from
forced precession of a magnetar by a companion or fallback disk, but is
consistent with theories where periodicity originates from a slowly-rotating or
freely-precessing magnetar. Several observational tests are proposed to
verify/differentiate between the remaining theories, and pin-down which theory
explains the periodicity in FRB 180916.
