An (oriented) matroid perspective (or morphism, or strong map, or quotient) is an ordered pair of (oriented) matroids satisfying some structural relationship. In this presentation, we will focus on the case of graphs, where two notable types of perspectives can be considered: graph homomorphisms, and dually embedded graphs on a surface. The Tutte polynomial of such a perspective is a classical polynomial (also called Las Vergnas polynomial in the case of dually embedded graphs), whose coefficients and (some) evaluations are known to count pairs of orientations of certain types. In this presentation, we show how coefficients and (other) evaluations of the polynomial also count pairs of orientations of certain types where some edge orientations are fixed, as well as some equivalence classes of pairs of orientations of certain types. These properties appear when the edge set is linearly ordered. Dedicated to the memories of Claude Berge and Michel Las Vergnas, as Claude Berge used to be the thesis advisor of Michel Las Vergnas who used to be my own thesis advisor.