We revisit the classical notions of confluence and orthogonality from the perspective of rewrite systems equipped with a residuation, modelled as 1-algebras. The perspective allows to smoothly connect confluence and orthogonality to various fields, e.g. to (least) upper bounds in order theory, (least) common multiples in algebra, and (relative) pushouts in category theory. Taking inspiration from the connexions, we prove 3 confluence results: completeness of random descent for completeness, 3-confluence by 3-local confluence, and orthogonality by undercutting. We illustrate the results by examples from various fields, and show each provides a solution to the problem of the calissons.