We give clear algebraic sense to logical opposition by providing negation-free lattice-theoretic definitions for the concepts around the notorious square of opposition. These include contradiction, contrariety, subcontrariety, the square and the hexagon of opposition, etc. This constitutes a platform for an analysis of the mathematical properties of logical opposition. We also discuss several examples of squares of opposition arising from universal logics studies, including Boolean as well as less conventional non-Boolean squares. The latter kind arise from a very general study of negation and consistency within the context of many-valued consequence relations. This work is a tribute to Jean-Yves Béziau, friend and colleague, on the occasion of his 50th birthday.