  We explore relations between Higgs bundles that result from isogenies between
low-dimensional Lie groups, with special attention to the spectral data for the
Higgs bundles. We focus on isogenies onto $SO(4,C)$ and $SO(6,C)$ and their
split real forms. Using fiber products of spectral curves, we obtain directly
the desingularizations of the (necessarily singular) spectral curves associated
to orthogonal Higgs bundles. In the case of $SO(6,C)$ our construction can be
interpreted as a new description of Recillas' trigonal construction.
