I don't understand your diagram, nothing's labelled (mind, I'm running for the door right now, will take a better look later)
If you search for order-switching you should find some pre-made diagrams.
With simple mechanical switches you'll always have one order that can't be done.
For example, swapping A&B and then adding a third circuit to order swap, C:
A>B>C
B>A>C
C>A>B
C>B>A
but not
A>C>B
B>C>A
To get C in between, you could have A as the outlier:
A>B>C
A>C>B
B>C>A
C>B>A
but not
B>A>C
C>A>B
etc.