The Senate would/could not investigate the House.
They are equal.
Statements by a House member in the Senate may not have legislative immunity though.
I think the House would appoint members with ‘clean hands’ as Managers.
.shall in all Cases, except Treason, Felony and Breach of the Peace, be privileged from Arrest during their attendance at the Session of their Respective Houses, and in going to and from the same; and for any Speech or Debate in either House, they shall not be questioned in any other Place.