For a manifold M admitting a metric g and given a second order symmetric tensor T on M one can construct from g and (the trace-free part of) T a fourth order tensor E on M which is related in a one-to-one way with T and from which T may be readily obtained algebraically. In the case when dimM = 4 this leads to an interesting relationship between the Jordan-Segre algebraic classification of T, viewed as a linear map on the tangent space to M with respect to g, and the Jordan-Segre classification of E, viewed as a linear map on the 6−dimensional vector space of 2−forms to itself (with respect to the usual metric on 2−forms). This paper explores this relationship for each of the three possible signatures for g.