Anyway, I think a left turn is incorrect. As previously mentioned, you must keep your turns on the side of the hold, so the only option as I see it, is a right turn until you're on an intercept course for the initial approach segment.
That doesn't seem consistent with what you write above. It is, as you say, just an initial approach segment. Why is there a requirement to "keep your turns on the side of the hold"? After passing TGO inbound for the final time, the hold is irrelevant.
The REUTL FOUR ALPHA is inbound to TGO on R163
which is a 90 degree left turn on reaching TGO. If you're prepared to make a 90 degree turn transitioning at the IAF from STAR to initial approach segment, why wouldn't you make a 154 degree turn from the hold to the initial approach segment?