The twisted odd graphs are obtained from the well-known odd graphs through an involutive
automorphism. As expected, the twisted odd graphs share some of the interesting properties
of the odd graphs but, in general, they seem to have a more involved structure. Here we
study some of their basic properties, such as their automorphism group, diameter, and
spectrum. They turn out to be examples of the so-called boundary graphs, which are
graphs satisfying an extremal property that arises from a bound for the diameter of a
graph in terms of its distinct eigenvalues.