I couldn’t find anything on the traditional workers party that stated they embraced socialism, and citing the their leader, Heimbach, as a second source is like counting nazi belief and Hitler’s belief as two cases, where they are in fact one.

As for Mussolini and Franco, they both had socialist views.

To me, I don’t see this as “right and left use socialism”, as much as I see facism as a hybrid of left and right beliefs. Socialism is a firm left wing position both politically and economically.