There's a stereotype that former Nazis often ended up being employed in East German security services, presumably as the required skill sets were overlapping. But how far was this actually true in reality? How was it perceived or justified internally?

I'm not so sure on how true this statement this, but can say a bit about how integration of former Nazis particularly in the military, bureaucratic and political spheres was justified. In both Germany's in the immediate aftermath of WW2 the Nazi regime was considered officially to have been a regime predominantly made up of a small clique of fanatics who repressed the German people - Germans were presented as much as victims as the people conquered by the Nazis, as the Nazi's rather then Germans were blamed for all that had gone wrong since 1933. There were differences in this broad narrative - west Germany presented the "victims" predominantly through the lenses of Germans who had been expelled from parts of central and eastern Europe during the Soviet advance and put primary responsibility of Nazi crimes on members of the Nazi party themselves (helping perpetuate the clean Wehrmacht myth amongst others). In East Germany, it was presented as the German workers' being repressed by a cabal of military officers, Nazi party functionaries, conservative bureaucrats and wealthy industrialists/capitalists hence the slightly stronger public denazification in the east. Neither of the Germany's acknowledged that the Nazis had been more-or-less fairly popular in the early 1930's and accepted by a large proportion of Germans, and often sought to explain Nazi rule through the lenses of Hitler somehow "tricking" the German populace through brilliant rhetoric and a sound economic policy.

Because the German people were presented moreso as victims of the Nazis rather then enablers it was easy to say that former Nazis had been forced into/tricked into their roles, and so bore no guilt for the atrocities done under the Nazi's, especially so if they weren't members of organisations like the SS (of which many former SS officers attempted to bury that part of their past due to the stigma it carried). It's only really in the late 1960's that in west Germany a serious reflection of German guilt for the Nazi regime takes hold in society, with that process beginning in earnest in the east following the collapse of the Berlin wall.

Source: The Search for Reconciliation: Sino-Japanese and German-Polish Relations since World War II, Yinan He. Note that this process of blaming a small clique for the atrocities of WW2 also applies for Japan as well and additionally helps explain why former militarist bureaucrats, generals and politicians continued to exercise power in post-war Japan.