Denote by $\varphi$ the automorphism of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$.This induces a self-map $B\varphi$ of $BO(n)$, so it induces a self-map (actually an involution) $B\varphi ^*$ on $[X,BO(2n)]\cong Vect_{2n}(X)$ where $Vect_{2n}(X)$ is the set of isomorphic classes of $2n$-dimensional real vector bundles over the topological space $X$. Or in terms of charts and transition functions, given a bundle $E$ with trivializing neighborhoods $U_i$ and transition functions $g_{i,j}$, one can construct a bundle $B\varphi ^*(E)$ with same trivializing neighborhoods $U_i$ and transition functions $\varphi \circ g_{i,j}$.

Question: is there any geometric interpretation of this involution? For example, can we describe geometrically $B\varphi ^*(T(RP^4))$ where $T(RP^4)$ denotes the tangent bundle over $RP^4$?