This is exactly why I added the higher order mode spacing, so you could calculate the g parameter.  For TEM order N = n + m with spacing f_N, the overall cavity g parameter should be:

g = (cos( (f_N/f_FSR) * (\pi/N) ))^2

The label on the previous plat should really be f_N/FSR, not \omega_{10,01}

BUT, arbcav does not currently handle arbitrary ABCD matrices for the mirrors, so it's going to be slightly less accurate for our more complex flipped mirrors.  The affect would be bigger for a flipped PR3 than for a flipped PR2, because of the larger incidence angle, so arbcav will be a little more correct for our flipped PR2 only case (see below).

This is not correct.  Multiplying the RoC by -N would be a very large change.  For an arbitrary ABCD matrix:

R_eff = -2 / C

When the incident angle in non-zero:

tangential: R_eff = R_eff / cos(\theta)
sagittal:   R_eff = R_eff * cos(\theta)

For flipped PR2, with small 1.5 degree incident angle and RoC of -706 at HR:

M_t = M_s = [1.0000, 0.0131; -0.0028, 1.0000]
R_eff = 705.9

For flipped PR3, with large 41 degree incident angle and RoC of -700 at HR:

M_t = [1.0000, 0; 0.0038, 1.0000]
M_s = [1.0000, 0; 0.0022, 1.0000]
R_eff = 592.4

The affect of the substrate is negligible for flipped PR2 but significant for flipped PR3.

The current half-PRC setup

OK, I have now completely reconciled my alamode and arbcav calculations.  I found a small bug in how I was calculating the ABCD matrix for non-flipped TTs that made a small difference.  I now get the exact same g parameter values with both with identical input parameters.

Sooooo, I have redone my alamode and arbcav calculations with these updated values.  Here are the resulting g parameters

So the sagittal values all agree pretty well, but the tangential measurement does not.  Maybe there is an actual astigmatism in one of the optics, not due to angle of incidence?

arbcav HOM plot: