The following preprint, exposed in my talk at CT2008, Calais, can be downloaded from my web page "The role of symmetries in cubical sets and cubical categories (On cubical categories, I)" (34 pages) http://www.dima.unige.it/~grandis/CCat.ps http://www.dima.unige.it/~grandis/CCat.pdf With best wishes Marco Grandis ______________ Abstract. Symmetric weak cubical categories were introduced in previous papers, as a basis for the study of cubical cospans in Algebraic Topology and higher cobordism. Such cubical structures are equipped with an action of the n-dimensional symmetric group on the n- dimensional component, which has been used to simplify the coherence conditions of the weak case. We give now a deeper study of the role of symmetries. The category of ordinary cubical sets has a Kan tensor product, which is non symmetric and biclosed, with left and right internal homs based on the right and left path functors. On the other hand, symmetric cubical sets have one path functor leading to one internal hom and a symmetric monoidal closed structure. Similar facts hold for cubical and symmetric cubical categories, and should play a relevant role in the sequel, the study of limits and adjunctions in these higher dimensional categories. Weak double categories, studied in four papers with R. Pare, are a cubical truncation of the present structures. While constructing examples of cubical categories, we also investigate a 'rewriting' procedure of reduction to canonical forms, which allows one to quotient a weak symmetric cubical category of cubical spans (resp. cospans), and obtain a strict symmetric cubical category of 'cubical relations' (resp. 'cubical profunctors'). ______________