  Most viable models of Type Ia supernovae (SN~Ia) require the thermonuclear
explosion of a carbon/oxygen white dwarf that has evolved in a binary system.
Rotation could be an important aspect of any model for SN~Ia, whether single or
double degenerate, with the white dwarf mass at, below, or above the
Chandrasekhar limit. {\sl Differential rotation} is specifically invoked in
attempts to account for the apparent excess mass in the super--Chandrasekhar
events. Some earlier work has suggested that only uniform rotation is
consistent with the expected mechanisms of angular momentum transport in white
dwarfs, while others have found pronounced differential rotation. We show that
if the baroclinic instability is active in degenerate matter and the effects of
magnetic fields are neglected, both nearly-uniform and strongly-differential
rotation are possible. We classify rotation regimes in terms of the Richardson
number, Ri. At small values of Ri $\leq$ 0.1, we find both the low-viscosity
Zahn regime with a non-monotonic angular velocity profile and a new
differential rotation regime for which the viscosity is high and scales
linearly with the shear, $\sigma$. Employment of Kelvin-Helmholtz viscosity
alone yields differential rotation. Large values of Ri $\gg$ 1 produce a regime
of nearly-uniform rotation for which the baroclinic viscosity is of
intermediate value and scales as $\sigma^3$. We discuss the gap in
understanding of the behavior at intermediate values of Ri and how observations
may constrain the rotation regimes attained by nature.
