  Galaxy angular momentum directions (spins) are observable, well described by
the Lagrangian tidal torque theory, and proposed to probe the primordial
universe. They trace the spins of dark matter halos, and are indicators of
protohalos properties in Lagrangian space. We define a Lagrangian spin
parameter and tidal twist parameters and quantify their influence on the spin
conservation and predictability in the spin mode reconstruction in $N$-body
simulations. We conclude that protohalos in a more tidal twisting environments
are preferentially more rotation-supported, and more likely to conserve their
spin direction through the cosmic evolution. These tidal environments and spin
magnitudes are predictable by a density reconstruction in Lagrangian space, and
such predictions can improve the correlation between galaxy spins and the
initial conditions in the study of constraining the primordial universe by spin
mode reconstruction.
