I have discussed relations between Italian Fascism and the various "conservative" forces of the Italian political landscape in a rather old post. There are a few things that I would re-arrange in that answer but overall it provides a decent summary.

Fascism was a political movement that rose from the area of "national radicalism" that involved groups such as the nationalists, veterans, futurists, etc. that converged for a brief time over a platform of "authoritarian renovation" of society (such as D'Annunzio's legionnaire movement). What allowed Fascism to build a more lasting consensus platform was arguably its ability to coalesce a large part of those fragmented groups and secure the support of those forces that were growing unsure of the liberal-conservative establishment. Thus Fascism was conservative in those areas where it had to meet the demands of its support basin; while retaining a large portion of the radical-progressive clothing of its early days.

But, as long as we focus on Italy, there was no "Conservative Party" - no political force that was openly concerned with a preservation of the status quo - to a point that it becomes hard to qualify some group as conservative without going into details.