Lattice path moments by cut and paste

In the coordinate plane consider those lattice paths whose step types consist of (1,1), (1,-1), and perhaps one or more horizontal steps. For the set of such paths running from (0,0) to (n+2,0) and remaining strictly elevated above the horizontal axis elsewhere, we define a zeroth moment (cardinality), a first moment (essentially, the total area), and a second moment, each in terms of the ordinates of the lattice points traced by the paths. We then establish a bijection relating these moments to the cardinalities of sets of selected marked unrestricted paths running from (0,0) to (n,0). Roughly, this bijection acts by cutting each elevated path into well-defined subpaths and then pasting the subpaths together in a specified order to form an unrestricted path.