  We define a bijection that transforms an alternating sign matrix A with one
-1 into a pair (N,E) where N is a (so called) ``neutral'' alternating sign
matrix (with one -1) and E is an integer. The bijection preserves the classical
parameters of Mills, Robbins and Rumsey as well as three new parameters
(including E). It translates vertical reflection of A into vertical reflection
of N. A hidden symmetry allows the interchange of E with one of the remaining
two new parameters. A second bijection transforms (N,E) into a configuration of
lattice paths called ``mixed configuration''.
