  We demonstrate a transversely polarized spin-exchange pumped noble gas
comagnetometer which suppresses systematic errors from longitudinal
polarization. Rb atoms as well as $^{131}$Xe and $^{129}$Xe nuclei are
simultaneously polarized perpendicular to a pulsed bias field. Both Xe
isotopes' nuclear magnetic resonance conditions are simultaneously satisfied by
frequency modulation of the pulse repetition rate. The Rb atoms detect the Xe
precession. We highlight the importance of magnetometer phase shifts when
performing comagnetometry. For detection of non-magnetic spin-dependent
interactions the sensing bandwidth is 1 Hz, the white-noise level is 7 $\mu$Hz
/$\sqrt{\text{Hz}}$, and the bias stability is $\approx1$ $\mu$Hz.
