  We study the spin- 1/2 two and three dimensional Orbital Compass Models
relevant to the problem of orbital ordering in transition metal oxides. We show
that these systems display self-dualities and novel (gauge-like) discrete
sliding symmetries. An important and surprising consequence is that these
models are dual to (seemingly unrelated) recently studied models of p+ip
superconducting arrays. The duality transformations are constructed by means of
a path-integral representation in discretized imaginary time and considering
its Z_{2} spatial reflection symmetries and space-time discrete rotations, we
obtain, in a transparent unified geometrical way, several dualities. We also
introduce an alternative construction of the duality transformations using
operator identities. We discuss the consequences of these dualities for the
order parameters and phase transitions of the orbital compass model and its
generalizations, and apply these ideas to a number of related systems.
