In the UK Parliament, there is a procedure whereby two MPs from opposite sides of the House can agree to both be absent for a vote, thereby preserving the same balance of votes as if both had attended. The pairing procedure can be used for all votes, except those indicated as a "three line whip" (for which all MPs are expected to vote if they are able)
The intention is to allow MPs to do constituency or government work, without being disturbed by numerous votes on relatively minor matters.
Is there a similar procedure used in the US House of Representatives or Senate?