PSP Tower Controller reported an aircraft on the Cathedral 1 Departure turned the wrong direction into conflict with another aircraft and indicated that this departure procedure routinely causes confusion for pilots due a lack of charted transition fixes.

Narrative

An IFR business jet departed Runway 31L on the Cathedral One Departure (runway heading for ~3 miles; procedural right turn to PSP VOR).Prior to departure; pilot had questioned Ground about the Cathedral One Departure as 1000s of pilots have done before; because of a systemic issue - I'll mention that in recommendations.On climb-out; I observed the pilot on runway heading (assigned course) and told him to contact Departure. After he read back the switch and was presumably talking to departure; I saw on the radar that aircraft was turning left; which is unsafe due to a 10000' mountain there.Even though he was presumably not on my frequency; I reached out for him 'callsign; if you are still on my frequency; you are supposed to be on runway heading until the 268 radial; runway heading.' there was no response. Aircraft continued left turn and was climbing straight into opposite direction VFR traffic at 4000. Aircraft was now climbing through 3200'. I issued a traffic alert just in case he was on my frequency. 'callsign; traffic alert; traffic 12 o'clock; 1 mile; opposite direction!' I failed to issue an advisory (stop your climb). Pilot never responded and was apparently receiving similar transmissions from Departure. The collision alert sounded on the radar. Afterwards; I called Departure and they told me that it was weird and the pilot just didn't seem to know what he was doing. I don't know if pilot was deviated or what came of it after that. The Cathedral One Departure is apparently a very confusing Departure because of it's requirement to climb to 8000. Pilots question the Departure procedure more than once on a daily basis. It is a non-radar SID that is built for slow climbers; and the fast climbers (all regular jets) are the ones that question it. Additionally; the CATH1 has no transitions; and pilots constantly question what to do after the Cathedral One (how do I continue on my route after completing the procedure.Another large problem we have with the Cathedral One is it defaults in many business jets FMS with the opposite direction runway (Runway 13R/L). Many business jet pilots tell us it doesn't connect to Runway 31L and we are forced to issue our only other SID which is not preferred in the letter of agreement (LOA). We never use the Cathedral One off of Runway 13R/L.I recommend the Cathedral One be corrected in whatever FMS company that default connects it to the wrong runway.This all being said; I understand we have some new departure procedures coming Palm Springs soon; so the other suggestion would be to no longer use the Cathedral One in relation to takeoff of Runway 31L/R and no longer have it the preferred departure procedure in the LOA.