BE400 Captain reported HSI; compass; transponder; and communications equipment failure resulted in a return to departure airport.

Narrative

Our clearance was 'After takeoff turn to 335 degrees; climb to 5000 feet. After takeoff; tower told us to turn to 335 degrees. I started the turn and noticed that my HSI (Horizontal Situation Indicator) was not turning. Informed PNF (Pilot Not Flying) that my HSI was not working; and I was going to use his (which looked correct). Tower then said they did not have our transponder. They then sent us over to ZZZ Approach. PNF monitored the turn and confirmed our heading of 335 degrees. PNF attempted to contact Approach 3 times with no answer. I then tried and was able to get them. During this time we leveled at 5000 feet. Approach said they did not have us on radar. PNF switched transponders; but approach still was not getting a target. Approach asked us our position. I said a few visual waypoints. He didn't know where they were; so I said '2 miles South of ZZZ1'. That's when we notice that the #2 HSI was not indicating correctly. A quick check of switches; circuit breakers and inverters revealed nothing. We agreed that we needed to return to ZZZ. We informed Approach that we were having avionics problems and needed to return to ZZZ. I informed ATC that we were below the Class B airspace; so we were canceling IFR and returning to ZZZ. ZZZ was in sight and ATC cleared us for a Visual approach to runway XX. We discovered the PNF com was working and ZZZ cleared us to land. Upon taxi in; neither HSI was showing the correct heading."