  We introduce and describe the "regular quotient" for the Hitchin fibration
for symmetric spaces and explain some basic consequences for Higgs bundles. We
include an invariant theoretic approach to spectral covers in this setting for
the particular space $\mathrm{GL}_{2n}/(\mathrm{GL}_n\times \mathrm{GL}_n)$. We
also include a study of the regular centralizer group scheme for quasisplit
pairs, including a Galois description of a closely related group scheme. We
collect some basic consequences for Hitchin systems associated to such pairs.
