  We construct supersymmetric models of SO(10) unification in which the gauge
symmetry is broken by orbifold compactification. We find that using boundary
conditions to break the gauge symmetry down to $SU(3)_C \otimes SU(2)_L \otimes
U(1)_Y \otimes U(1)_X$ without leaving unwanted massless states requires at
least two extra dimensions, motivating us to work with 6D orbifolds. SO(10) is
broken by two operations, each of which induces gauge-breaking to either the
Georgi-Glashow, Pati-Salam, or flipped $SU(5) \otimes U(1)$ subgroups;
assigning different unbroken subgroups to the two operations leaves only the
standard model gauge group and $U(1)_X$ unbroken. The models we build employ
extra-dimensional mechanisms for naturally realizing doublet-triplet splitting,
suppressing proton decay, and avoiding unwanted grand-unified fermion mass
relations. We find some tension between being free of anomalies of the 6D bulk,
accommodating a simple mechanism for generating right-handed neutrino masses,
and preserving the precise prediction of the weak mixing angle.
