  We consider incommensurate $d$-density wave order in underdoped high
temperature superconductors. We find that Fermi surface reconstruction can
correctly capture the phenomenology of the recent quantum oscillation
experiments that suggest incommensurate order. The predicted frequencies are a
frequency around 530 T arising from the electron pocket, a hole frequency at
around 1650 T, and a new low frequency from a smaller hole pocket at 250 T for
which there are some indications that require further investigation. The
oscillation corresponding to the electron pocket will be further split due to
bilayer coupling but the splitting is sufficiently small to require more
refined measurements. The truly incommensurate $d$-density wave breaks both
time reversal and inversion but the product of these two symmetry operations is
preserved. There is some similarity of our results with the spiral spin density
wave order, which, as pointed out by Overhauser, also breaks time reversal and
inversion. Calculations corresponding to higher order commensuration produces
results similar to anti-phase spin stripes, but appear to us to be an unlikely
explanation of the experiments. The analysis of the Gorkov equation in the
mixed state shows that the oscillation frequencies are unshifted from the
putative normal state and the additional Dingle factor arising from the
presence of the mixed state can provide a subtle distinction between the spiral
spin density wave and the $d$-density wave.
