Abstract : The spin properties of the evanescent states in a semiconductor with no inversion center are studied. The peculiar topology of the evanescent bands which results from the spin-orbit interaction is at the origin of an anomalous behavior of the tunneling. Even the nature of the tunnel process itself strongly depends on the crystallographic orientation. Two typical cases are analyzed: tunneling under off-normal incidence through a -oriented barrier and tunneling under normal incidence through a -oriented barrier. In the first case, a quasi-classical process can be restored in a quite subtle way and spin-filter effects are evidenced. In the latter case, the situation is completely unusual. The notion of probability current, which plays a central role, is re-examined and conditions expressing the discontinuity of the wave-function derivative at the interfaces are derived. Strangely enough, in this direction where the spin-splitting of the real conduction band is maximum, there is no spin-filter effect. The transmitted wave experiences a phase shift with a sign depending on the spin orientation. Spin-precession phenomena around the complex internal field existing in the barrier are predicted. Making use of spin-orbit engineering of heterostructures, these results allow one to conceive resonant-tunnel devices with spin-manipulation capability.