  In this article, we prove a new general identity involving the Theta
operators introduced by the first author and his collaborators in [D'Adderio,
Iraci, Vanden Wyngaerd 2020]. From this result, we can easily deduce several
new identities that have combinatorial consequences in the study of Macdonald
polynomials and diagonal coinvariants. In particular, we provide a unifying
framework from which we recover many identities scattered in the literature,
often resulting in drastically shorter proofs.
