  We investigate the spin-orbit (s-o) interaction in two-dimensional electron
gases (2DEGs) in quantum wells with two subbands. From the $8\times 8$ Kane
model, we derive a new inter-subband-induced s-o term which resembles the
functional form of the Rashba s-o -- but is non-zero even in \emph{symmetric}
structures. This follows from the distinct parity of the confined states
(even/odd) which obliterates the need for asymmetric potentials. We
self-consistently calculate the new s-o coupling strength for realistic wells
and find it comparable to the usual Rashba constant. Our new s-o term gives
rise to a non-zero ballistic spin-Hall conductivity, which changes sign as a
function of the Fermi energy ($\epsilon_F$), and can induce an unusual
\emph{zitterbewegung} with cycloidal trajectories \textit{without} magnetic
fields.
