object clause

[~Mav~]

To show our respect, we take our gloves off whoever we are to shake hands with.

I think it would be correct with a comma between "off" and "whoever".

That was my very first thought! (In fact, I had automatically read the sentence that way, and only on second read did I realise what this thread was about. )

In this case, "whomever" would be impossible.

Unless - in the highly unlikely case - one says, "To show our respect, we take our gloves off, with whomever we are to shake hands." (I know this is not what you meant. )

PS: Maybe a slightly more natural sentence would be, "We take our gloves off to show our respect, with whomever we are to shake hands."