if only the relative order around the table is of importance...
I assume that we do not need to eliminate a rotation factor
First, if "only the relative order around the table is of importance" then you must surely eliminate the rotation factors.
Second, even if your assumption was correct, your answer would be wrong. You have only considered the number of permutations should the women be in the odd (or even) position and the men in the other, even (or odd) positions. You need to multiply this by 2 in order to get them all.
Then if you need to eliminate the rotations, divide by 10. You will end up with 5! x 4!, the same result you would get if you were to pick one person as an anchor.
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993