  We examine the anomalous inverse spin switch behavior in
La$_{0.7}$Ca$_{0.3}$MnO$_3$ (LCMO)/YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO)/LCMO
trilayers by combined transport studies and polarized neutron reflectometry.
Measuring magnetization profiles and magnetoresistance in an in-plane rotating
magnetic field, we prove that, contrary to many accepted theoretical scenarios,
the relative orientation between the two LCMO's magnetizations is not
sufficient to determine the magnetoresistance. Rather the field dependence of
magnetoresistance is explained by the interplay between the applied magnetic
field and the (exponential tail of the) induced exchange field in YBCO, the
latter originating from the electronic reconstruction at the LCMO/YBCO
interfaces.
