A TRACON Controller reported a departing aircraft turned to the wrong heading and flew below the Minimum Vectoring Altitude.

Narrative

Aircraft X was departing ZZZ1 to ZZZ. ZZZ1 is located 2NM north of ZZZ and requested IFR opposite direction departure off RWY X at ZZZ1 from the ZZZ Local Controller. The Local Controller issued a heading of 360 maintain 8000 Ft. to the ZZZ1 LC (Local Control). Aircraft X then departed ZZZ1 and checked in with me on the Re XXX.XX Freq. They stated 'Aircraft X departing ZZZ1 climbing to 8000 heading 260.' I initially did not hear the 260 heading as it was garbled and we currently had frequency issues on several frequencies. I verified that Aircraft X was indeed inbound to ZZZ and after they confirmed I issued a 6500 ft. altitude and to expect visual approach to Rwy XX at ZZZ. The aircraft was turning pretty quickly and looked as though they were turning past a 360 heading so I assigned heading 360 again. The aircraft readback heading 360. The aircraft was indeed turning beyond 360 and the aircraft's momentum took it into a 8000 and 8500 Ft. MVA west of ZZZ. As soon as I saw it making a westbound turn I issued an immediate heading of 020 and then a low altitude safety alert and a further turn to 060. The aircraft was then vectored to Rwy XX at ZZZ. The pilot was given a Brasher Warning after landing and after questioning the pilot to why they turned westbound; they replied that ZZZ1 had issued a heading of 260; climb and maintain 8000. An ODO (Opposite Direction Operation) departure request such as this one is extremely common between ZZZ1 and ZZZ. The ZZZ1 LC apparently issued a 360 maintain 8000 instruction but missed a 260 maintain 8000 readback from Aircraft X. The controllers at ZZZ1 should excercise the same practices as we do at FAA facilities and have some sort of direct supervision in the tower cab to assist in catching these types of errors. The pilot of Aircraft X stated that they were familiar with the area and the high terrain yet he did not question a westbound heading into a mountain range 5 to 7 NM west of our airports. I believe there was a lack of attention to detail on behalf of Aircraft X; ZZZ1 and ourselves at ZZZ.