Igor, I think conventionally, the "default" order on \\(X \times X\\) is the product (component-wise) order. When we write a monotone function \\(f : X \times X \to X\\), we're leaving the orders implicit for both \\(X \times X\\) and \\(X\\), but there's still a _fixed_ order for both, which all functions in \\(M^\times\\) must respect. In that sense, \\(C^\times\\) is indeed the defining characteristic of \\(M^\times\\).
Put another way, a function \\(f : X \times X \to X\\) may be monotone with respect to the product order, but not with respect to the lexicographical order -- or vice versa. Thus, \\(M^\times\\) _can't_ be defined independent of any particular order.