Inspire by this, I wonder
Pontrjagin square: There is a geometric interpretation of $\mathfrak{P}_2$, due to Morita.
Assume $q=2k$, so that the Pontrjagin square is a map $$\mathfrak{P}_2 \colon H^{2k}(X, \mathbb{Z}_2) \longrightarrow \mathbb{Z}_4.$$ Set $$Z \equiv \sum_{t \in H^{2k}(X, \mathbb{Z}_2)} e^{2 \pi i \mathfrak{P}_2(t)}.$$ Then
$$\textrm{Arg}(Z)= \textrm{Arg}( \sum_{t \in H^{2k}(X, \mathbb{Z}_2)} e^{2 \pi i \mathfrak{P}_2(t)})= \frac{\sigma(M)}{8} \in \mathbb{Q}/\mathbb{Z}, \tag{(a)}$$
where $\sigma(X)$ denotes the signature of $M$.
For a proof, see Gauss Sums in Algebra and Topology, by Laurence R. Taylor.
Postnikov square:
Question: Are there similar statements for Postnikov square? (like eqn.(a))
Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, introduced by Postnikov (1949). Eilenberg (1952) described a generalization taking classes in $H^t$ to $H^{2t+1}$.
Refs:
Postnikov, M. M. (1949), "The classification of continuous mappings of a three-dimensional polyhedron into a simply connected polyhedron of arbitrary dimension", Doklady Akademii Nauk SSSR (N.S.) (in Russian), 64: 461–462
Eilenberg, Samuel (1952), "Homotopy groups and algebraic homology theories", Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, 1950, vol. 2, Providence, R.I.: Amer. Math. Soc., pp. 350–353, MR 0045388