It is a classical result of F. Klein that for any nonorientable (regular enough) surface $\boldkey X $ there is an orientable surface ${\Cal O}_2$ and an involution without fixed point of ${\Cal O}_2$ such that $\boldkey X $ is isomorphic to the quotient space of ${\Cal O}_2$ with respect to the group generated by the respective involution. In this note a reinforcement of the Klein's result is presented and the effect on the vector bundle of covariant tensors of second order on X produced by that involution is studied. The projection $p:{\Cal O}_2 \longto \boldkey X $ induces an isomorphism between the vector space of covariant tensors of order two on $\boldkey X$ and the space of covariant symmetric tensors of order two on ${\Cal O}_2$ which are invariant with respect to the given involution.