If the boundary of a manifold has singularities, e.g., conical points or edges, then when dealing with boundary value problems, one may be forced to work with pseudo-differential operators with corner degenerated symbols. In this paper, elements of the corresponding corner pseudo-differential calculus are studied. In particular, the authors introduce Green symbols that play the role of specific operator-valued symbols in the calculus, and also operator-valued Mellin symbols. Moreover, they study the non-smoothing elements of the corner calculus, corner-degenerated pseudo-differential symbols, and construct various quantisations in the form of operator-valued symbols with twisted homogeneity.