Yes your two LB'S would be aligned on the equinox ,and with the same orientation if the horizon for each was similar all that would be required of the builders, is to know when the equinox took place , a matter of counting of days from the solstice .

But despite the similarities you might only get get three or four close to that exact alignment ,and there is leeway , maybe about 30 degrees .

The counting of days , might explain some similar orientations , i.e. group A decides to orient their monuments on the rising sun 90 days after the winter solstice meanwhile elsewhere group B chooses 111 days .

It seems a bit odd/unlikely but it would work . But if the orientation is towards a part of the sky where the sun never rises or sets ,which is often the case ,and that quarter of the horizon could encompass 100 degrees or more then counting won't help .