Let A, B be standard operator algebras on complex Banach spaces X and Y of dimensions at least 3, respectively. In this paper we give the general form of a surjective (not assumed to be linear or unital) map Φ : A → B such that Φ2 : M2 (C) ⊗ A → M2 (C) ⊗ B defined by Φ2 ((s ij) 2×2) = (Φ(s ij)) 2×2 preserves nonzero idempotency of Jordan product of two operators in both directions. We also consider another specific kinds of products of operators, including usual product, Jordan semi-triple product and Jordan triple product. In either of these cases it turns out that Φ is a scalar multiple of either an isomorphism or a conjugate isomorphism.