  General relativity predicts that two counter-orbiting clocks around a
spinning mass differ in the time required to complete the same orbit. The
difference in these two values for the orbital period is generally referred to
as the gravito-magnetic (GM) clock effect. It has been proposed to measure the
GM clock effect using atomic clocks carried by satellites in prograde and
retrograde orbits around the Earth. The precision and stability required for
satellites to accurately perform this measurement remains a challenge for
current instrumentation. One of the most accurate clocks in the Universe is a
millisecond pulsar, which emits periodic radio pulses with high stability.
Timing of the pulsed signals from millisecond pulsars has proven to be very
successful in testing predictions of general relativity and the GM clock effect
is potentially measurable in binary systems. In this work we derive the generic
GM clock effect by considering a slowly-spinning binary system on an elliptical
orbit, with both arbitrary mass ratio and arbitrary spin orientations. The
spin-orbit interaction introduces a perturbation to the orbit, causing the
orbital plane to precess and nutate. We identify several different
contributions to the clock effects: the choice of spin supplementary condition
and the observer-dependent definition of a full revolution and
"nearly-identical" orbits. We discuss the impact of these subtle definitions on
the formula for GM clock effects and show that most of the existing formulae in
the literature can be recovered under appropriate assumptions.
