my approach would have been different : active rectifier followed by a single comparator.
the advantage of the active rectifier is that the triggerpoint will be symmetrical around the reference point.
and i dont need a negative rail either...
there is only 1 comparator doing the work.
hows this work ?
well, during the positive half of the period:
ic1 a output will follow input ( he's basically a unity gain buffer )
ic1b output sees the signal from ic1a and the input signal. that thing is set up for a gain of 2 (r2=r3). so it subtracts VO from 2xVin yielding 1xVin.
during the negative half of the period:
ic1a locks up ( it cant measure below ground ... since it has no 'negative supply) so V0 becomes effectively ground.
IC2 a will produce a voltage to make its - input also be '0'. if vin is -1 volt vout will be +1 volt. since r2=r3 the system is in balance. so for negative signals this thing runs basically unity gain for positive signals it runs a gain of 2 but subtracts 1 vin yielding again the net effect of unity gain.
there you go. an ideal rectifier with only 1 power rail. slap on a comparator and you are good.
r1 is needed so you dont fry the esd diode in ic1 when you pull the input below ground.
in case you have a DC offset at the input : add a coupling capacitor.