Forward/reverse median filter. Reference needed
I've held off on answering this because I don't have MATLAB to test any of this.

I've employed the same trick with exponential smoothing.  The purpose of this is to have the lags in the opposite direction cancel each other out, so the resultant smoothed signal is centered instead of shifted from the source signal.  For sinusoids, the lag cancellation is exact.

So, I thought this might be the same motivation in this.  It does not seem to be the case.  (Side note:  In my searching I found the MATLAB function "movmedian".  I don't immediately see how it is that different from "medfilt1".)

I agree with Laurent Duval that if an odd number is used that the forward and backward filtered signals will be the same.  However, OP is using an even number.  It seems to me that the backward filtered signal will be just one index off of the forward one since the window size is the same and the median for each window should give the same result.  Thus, excepting the end cases, you should be able to just do the forward one and then define:

$$ M[x] = ( Mfw[x] + Mfw[x+1] ) / 2 $$

Which is a very simple FIR.

Two cents minus inflation worth.

Ced