I was not able of reproducing your results. Let me show you my trial code.
First, I assign some basic properties:
Then I check if the inverse metric is symmetric:
I'm getting that it is
Just as a cross check I ask if it is antisymmetric:
And I got
If you are not getting the same results it might be that you are using an older version of the software, were the inheritance of the symmetries was buggy or miscompiled.