Sufficient conditions are obtained for the continuity of renormalized self-intersection local times for the multiple intersections of a large class of strongly symmetric L vy processes in $R DEGREESm$, $m=1,2$. In $R DEGREES2$ these include Brownian motion and stable processes of index greater than 3/2, as well as many processes in their domains of attraction. In $R DEGREES1$ these include stable processes of index $3/4