Let $X$ be a manifold and $E\to X$ a complex vector bundle and let's work in $H^\bullet(X,\mathbb{Z})$. Given the total Chern class of $E$, $c(E)=1+c_1(E)+\cdots+c_n(E)$, we can define the total Segre class of $E$ to be $s(E)=1+s_1(E)+\cdots+s_n(E)$ to be the inverse of the total Chern class.
Equivalently, recalling that $c(E)=\prod_{i=1}^n1+\alpha_i(E)$, with $\alpha_i(E)$ the $i$-th Chern root of $E$, we can define $$s(E)=\prod_{i=1}^n\frac{1}{1+\alpha_i(E)}=\prod_{i=1}^n\sum_{k=0}^\infty (-\alpha_i(E))^k.$$
I am interested in the real analogue, now working in $H^\bullet(X,\mathbb{F}_2)$, that is to say the following:
given $V\to X$ a real vector bundle, the total Stiefel-Whitney class of $V$ is $$w(V)=1+w_1(V)+\cdots+w_n(V)=\prod_{i=1}^n1+\sigma_i(V)$$ (where the $\sigma_i(V)$ are the Stiefel-Whitney roots of $V$) and we can analogously define its inverse as $$p(V)=\prod_{i=1}^n\frac{1}{1+\sigma_i(V)}=\prod_{i=1}^n\sum_{k=0}^\infty\sigma_i(V)^k=1+p_1(V)+\cdots+p_n(V).$$
This all seems quite natural but I have never seen it anywhere, the only sources I could find are [1] Stiefel-Whitney Homology Classes and Riemann-Roch Formula of Matsui and Sato and [2] Axioms for Stiefel-Whitney Homology Classes of Some Singular Spaces of Veljan, but they work in a different, more general context.
To conclude, I'm asking if these classes have been studied and what else is known about them. My main concern is the integral of $p_n(V)$ with [1] and [2] suggesting that $\int_Xp_n(V)=\chi(V)\operatorname{mod} 2$, but I would appreciate more sources.
