Let {Rn($chi$)}{{{{ { } atop {n=0} }}}} be a discrete Sobolev orthogonal polynomials (DSOPS) relative to a symmetric bilinear form (p,q)={{{{ INT _{ } }}}} pqd$mu$0 +{{{{ INT _{ } }}}} p qd$mu$1, where d$mu$0 and d$mu$1 are signed Borel measures on . We find necessary and sufficient conditions for {Rn($chi$)}{{{{ { } atop {n=0} }}}} to satisfy a second order difference equation 2($chi$) y($chi$)+ 1($chi$) y($chi$)= ny($chi$) and classify all such {Rn($chi$)}{{{{ { } atop {n=0} }}}}. Here, and are forward and backward difference operators defined by f($chi$) = f($chi$+1) - f($chi$) and f($chi$) = f($chi$) - f($chi$-1).