Single-particle dynamics in simple models for the static reversing magnetic field of the geotail current sheet have been extensively studied in the case where the reversing field Bx(z) varies, the linking field Bz is constant, and the crosstail field By is zero (and some generalization to include time dependence has been achieved). More recently, numerically integrated trajectories in static reversals which include a constant shear By component have suggested some differences in the nature of the dynamics in this and the By=0 case. The invariant of the z cross-sheet motion for the By=0 case is well known, here we find its equivalent for systems with constant By. Our results hold for reversals with general z dependence and arbitrary constant By. The form of this invariant suggests that it is still conserved for trapped particles, but for certain values of energy, By and Bz, the invariant is destroyed and particles are detrapped. This corresponds to an increase in the volume in phase space available to current carrying particles that transit the sheet. For typical magnetotail parameters, both protons and electrons in an average 1 RE thick sheet will be detrapped, but in a thin ~100 km sheet protons will not. ¿ 1998 American Geophysical Union