A method, comprising providing a downhole electromagnetic logging tool having two or more antenna groups spaced along a longitudinal axis of the logging tool, each antenna group comprising: one antenna (T1, T2, T3, R1, R2) or multiple antennas (T1, T2, T3, R1, R2) proximately located or collocated; and at least one antenna (T1, T2, T3, R1, R2) having a dipole moment that is tilted relative to the longitudinal axis of the logging tool, wherein at least one antenna group having, in addition to the at least one antenna having a dipole moment that is tilted relative to the longitudinal axis of the logging tool, at least one antenna (T1, T2, T3, R1, R2) having a dipole moment that is transverse or tilted relative to the longitudinal axis of the logging tool, and any given pair of antenna groups comprising at least four antennas (T1, T2, T3, R1, R2) with at most three of those antennas having dipole moments that are tilted relative to the longitudinal axis of the logging tool; the method further comprising disposing the logging tool in a wellbore penetrating a formation; and rotating the logging tool; the method further comprising, using a processor to perform the steps of: measuring a tool rotation angle while rotating the logging tool; transmitting an electromagnetic signal from one antenna group while rotating the logging tool; receiving the electromagnetic signal with another antenna group while rotating the logging tool; obtaining, from the received electromagnetic signal and the measured tool rotation angle, sufficient information to determine all of the elements of an electromagnetic coupling tensor; determining some or all of the elements of the electromagnetic coupling tensor using the obtained information; and inferring petrophysical properties of the formation using one or more elements of the determined electromagnetic coupling tensor. providing a downhole electromagnetic logging tool having two or more antenna groups spaced along a longitudinal axis of the logging tool, each antenna group comprising: one antenna (T1, T2, T3, R1, R2) or multiple antennas (T1, T2, T3, R1, R2) proximately located or collocated; and at least one antenna (T1, T2, T3, R1, R2) having a dipole moment that is tilted relative to the longitudinal axis of the logging tool, one antenna (T1, T2, T3, R1, R2) or multiple antennas (T1, T2, T3, R1, R2) proximately located or collocated; and at least one antenna (T1, T2, T3, R1, R2) having a dipole moment that is tilted relative to the longitudinal axis of the logging tool, wherein at least one antenna group having, in addition to the at least one antenna having a dipole moment that is tilted relative to the longitudinal axis of the logging tool, at least one antenna (T1, T2, T3, R1, R2) having a dipole moment that is transverse or tilted relative to the longitudinal axis of the logging tool, and any given pair of antenna groups comprising at least four antennas (T1, T2, T3, R1, R2) with at most three of those antennas having dipole moments that are tilted relative to the longitudinal axis of the logging tool; the method further comprising disposing the logging tool in a wellbore penetrating a formation; and rotating the logging tool; the method further comprising, using a processor to perform the steps of: measuring a tool rotation angle while rotating the logging tool; transmitting an electromagnetic signal from one antenna group while rotating the logging tool; receiving the electromagnetic signal with another antenna group while rotating the logging tool; obtaining, from the received electromagnetic signal and the measured tool rotation angle, sufficient information to determine all of the elements of an electromagnetic coupling tensor; determining some or all of the elements of the electromagnetic coupling tensor using the obtained information; and inferring petrophysical properties of the formation using one or more elements of the determined electromagnetic coupling tensor. measuring a tool rotation angle while rotating the logging tool; transmitting an electromagnetic signal from one antenna group while rotating the logging tool; receiving the electromagnetic signal with another antenna group while rotating the logging tool; obtaining, from the received electromagnetic signal and the measured tool rotation angle, sufficient information to determine all of the elements of an electromagnetic coupling tensor; determining some or all of the elements of the electromagnetic coupling tensor using the obtained information; and inferring petrophysical properties of the formation using one or more elements of the determined electromagnetic coupling tensor.