Cross-correlation with SDPSK modulation I'm trying to detect a sync word in a signal using Symmetrical Differential PSK (bit 0 = +90° phase, bit 1 = -90° phase).

Given the signal is encoded in the phase difference between two symbols, the absolute phase is irrelevant. How would I go about using cross-correlation to detect the sync word?

I have four possible orientations for the sequence considering the absolute phase, but for performance reasons, I would prefer not to run the cross-correlation more than once. Should I calculate the phase difference of each symbol with the previous (i.e. extract the bits) and correlate on that or is there a way to do it without first converting the signal to bits?
Convert your input signal to an analytic signal with a Hilbert transformer, if your input signal is real.  Then run a single complex cross correlation filter.  Assuming no frequency offset, the phase of the correlator output at the correlation peak will be the input signal's phase shift from your reference signal.