GA pilot reported being confused about the clearance from ATC after initial climb and was contacted by ATC of a low altitude alert and given instructions to turn immediately.

Narrative

Departed out of SMO on IFR clearance with 'fly runway heading to 1000 then right-hand turn to 250 expect radar vectors to V23....' I noted that V23 was only a few miles off the end of the runway. I departed; turned right; heading 250; and switched frequencies. I thought the controller said turn right 270 intercept V23. But I think he just said turn right 270. I was on top of V23 so I intercepted and was in a right turn to intercept (about 350). Less than a minute later; just as I was assessing if the turn made sense and was going to ask ATC to confirm the intercept instructions; the controller said; 'Low altitude alert turn left immediately to 270; which I did. Later was told to call ATC on the phone; which I did. I still don't know if I got the intercept clearance; but when I spoke to the controller; he said they did not give me an intercept.I probably should have queried immediately as I flew over V23. It was a very unusual clearance to begin with (that one would overfly the airway and then be vectored back to intercept; but I guess that is how the routing is designed. This can lead to the confusion that I had.