BAe 125 flight crew reported HSI failure during initial departure climb.

Narrative

On Day 0 from approximately XA30z to XA50z; Aircraft X experienced an air turn-back to ZZZ due to a failure of the #1&#2 HSI heading data/presentation. The turn back was uneventful and in compliance with the aircraft check- lists/procedures. During a left-seat; hand-flown departure in VMC from Runway X at approximately 400 ft; I initiated an ATC assigned heading to 060 degrees magnetic. During the turn; I observed the HSI precess to the right (ESE). As I held the bugged 060 heading; the aircraft continued its right bank. Simultaneously; ZZZ Tower radioed the 'I really need you on that 060 heading'. The First Officer (FO) replied that we were on a 060 heading. I directed the FO to communicate to ATC that we having a problem with our heading information and to request a vector back to the left.ATC issued a vector-heading to the left and I complied immediately. It was at this point I noticed that the FO's HSI was also precessing to the right at the same rate and magnitude. I compared the heading of #1&#2 HSI to the aircraft's standby compass and noted a difference of 90-120 degree. We relayed to ATC that we needed to return to ZZZ. ATC offered; and we accepted; no-gyro vectors back for a visual approach to Runway X. While maneuvering; ATC asked if were declaring an emergency; we said no; we were not. ATC further inquired as to souls and fuel on board and whether we needed Crash Fire Rescue (CFR) support after landing. We advised that we would be performing an over-weight landing as we were fueled for a planned extended flight. There were not any indications of heading mis-compare or reversion. All other navigation and aircraft systems operated as expected and provided normal indications. The remainder of the operation was routine; approach; landing; roll-out; and taxi were uneventful and in accordance with all applicable aircraft checklists.ATC advised that CFR would follow us to parking 'as a precaution'. After shutdown; I inspected the aircraft for obvious damage and reported to CFR personnel that the aircraft was undamaged and secure. I immediately contacted our Supervisor and reported the occurrence. The aircraft condition was reported to Maintenance and subsequently entered into the aircraft logbook. The entire incident took place in approximately 18 minutes of flight time. There were no indications prior to departure of any aircraft discrepancies. During taxi and runway line-up; both HSI's operated as expected and heading bugs set and verified to runway heading prior to applying take-off power. The precession did not occur until the aircraft was airborne.