In this paper we construct several infinite families of W-algebras as S₂-invariant subalgebras of the tensor product of two copies of affine vertex W-algebras of "rank" two. For every value of the central charge (or level) we completely determine their types in terms of strong generators. We also consider simple quotients in the case when the type is different from in the generic case. We also examine examples of conformal embeddings that come out of the construction.

A.M. was partially supported by NSF grant DMS-2101844.