For $\mathbb{Z}_2$ cohomology classes, we have a very useful Wu formula: In $d$-dimensional manifold and for a $n$-cocycle in $x_n \in H^n(M^d; \mathbb{Z}_2)$, we have $Sq^{d-n}(x_n)=u_{d-n}\cup x_n$, where $u_m$ is the $m^{th}$ Wu class, and $Sq^k$ is the Steenrod square.

I wonder if there is a similar formula for integral cohomology classes? For example, if there is a cohomological operation $P^4$, such that $P^4(y_{d-4}) = p_1 \cup y_{d-4}$ where $p_1$ is the first Pontryagin class and $y_n \in H^n(M^d; \mathbb{Z})$?