It's pretty clear that both the flight crew and the controllers have some blame in this situation, but for different reasons. Let's go through this bit by bit.
Firstly, the error on the initial incorrect left turn to 180 will fall onto the controller one way or another, either because she incorrectly spoke "left" instead of "right" or, if it turns out that she didn't mispeak and correctly cleared them "right" then she missed the incorrect readback of "left". So, no matter what, she screwed up on at least one of those two issues.
After noticing EVA turning left she gives a good proper instruction of a right turn to 180 to get them back on the correct heading south. Nothing wrong with that. Well done.
However, separation is quickly reducing between EVA and an Air Canada 767 that departed LAX after EVA. She instructs EVA to expedite the right turn, and the crew responds in the affirmative that they are in the right turn passing through 010 heading. Again, nothing is getting out of hand at this point. If EVA just continues their right turn around to 180, everything will turn out just fine. But this is where it starts going haywire.
She then starts to give a mistaken clearance to Air Canada 788 to expedite their turn and stop their climb, which makes no sense because they were never in a turn to begin with, they were on a steady heading to intercept their SID, but they were indeed in a climb so they could follow that instruction at least. She notices her error and corrects herself telling Air Canada to expedite their climb (instead of stopping it), and to turn left to 360, which would be a fine instruction at this point. HOWEVER, because of the botched initial instruction to stop the climb, Air Canada reads back that they will be stopping their climb at 7,000, along with the correct clearance to 360. They did not hear or understand the corrected instruction to expedite their climb, and she does not correct them, so they level off at 7,000.
She then tells EVA to stop their climb.
Both aircraft have now stopped their climb, and are roughly parallel with each other in roughly the same few miles of airspace. Air Canada on the left at 7,000, EVA on the right at 5,000 and still cleared to turn right to 180.
She now notices Air Canada's leveling off at 7,000 and tells them to expedite their climb to 12,000. This catches Air Canada off guard because of the hasty instruction to stop their climb just a few seconds ago, so they ask for confirmation of the climb up to 12,000. She confirms. Air Canada begins a steep climb up to 12,000.
She then starts to clear EVA for a left turn to 270 which is in conflict with the prior clearance to the right to 180. EVA acknowledges. This may have been due to a UPS aircraft climbing out of Ontario airport to the east of EVA, so the controller may have decided it was best to go around to the west instead. Also, This clearance may seem odd at first glance because she's giving EVA a turn back to the left when Air Canada is directly off their left, but at this point Air Canada is climbing and should not be a factor. EVA should pass well underneath Air Canada if they turn left onto 270. Either way, changing their clearance from right turn to left turn was a big change, and added to the confusion.
At this point EVA is now starting to turn left to the 270 heading they were cleared to. After a few moments she then asks EVA "What are you doing" and to "Turn southbound" along with "Stop your climb". First of all, they are not in a climb, and have been level at 5,000 for quite a few minutes at this point. Secondly, If the crew was slow getting the left turn started, along with the radar display lag, she may not have noticed they had begun their left turn, and believed they were still flying north. This is where things get completely lost on both sides of things. EVA likely gets confused because they are doing exactly what she said. They are turning left to 270. Now the controller is yelling at them "what are you doing". So they hesitate and stop their turn still heading roughly north. EVA asks for clarification, and stutters "left...right...heading" before stopping transmission. This clearly shows that they were confused at this point as to whether they were supposed to be turning left or right, and to what heading. Left would make sense, because they were given a left clearance to 270. But when she yelled at them and told them to turn south immediately, they likely second guessed their turn direction, not sure if they misunderstood something to warrant her reaction.
For foreign pilots, they were probably completely lost at this point. First they were told to correct with right turn to 180, then they were told to turn left to 270, so they started that left turn, then got yelled at and told to turn southbound (180). They were getting all sorts of conflicting clearances that weren't in standard ATC verbiage. They probably couldn't decide whether they were supposed to turn left or right. After all, they tried left before and got yelled at.
At this point, Air Canada is cleared to 12,000 ft, well above EVA and out of the picture.
EVA now requests confirmation on the heading, obviously still not sure what heading or direction the controller wants them to turn. They even seem to stress the world "heading", likely because they want an actual turn direction and HEADING, and not a rough direction like "south".
She responds with "Southbound, southbound now!", still not giving them a proper heading or turn direction.
The crew gives up on trying to get a heading, and reads back "roger, southbound now". They still have not clarified the turn direction, but they decide amongst themselves to make it a right turn, likely because of the admonishment they got last time when they turned left. They are still at 5,000 heading toward rising terrain that reaches around 6,000 feet.
After dealing with another aircraft, she then tells EVA to climb and maintain 5,000 (which they are already at), along with asking them "Are you southbound now, I see you going northbound" as well as changing (in the same transmission) their climb clearance, telling them to climb and maintain 6,000. EVA replies "southbound, maintain 5,000." missing her corrected 6,000 ft climb instruction. They are in a sweeping right hand turn to the right at this point, heading northeast toward Mt Wilson.
She then orders them to climb to 7,000. They readback correctly. They continue their turn and begin to climb.
She then says to them "I see you heading southbound now, turn south- correction, I see you heading northbound, turn southbound now". This constant mis-speaking and rapid fire corrections of non-proper ATC instructions is going to be absolutely confusing to a foreign flight crew who is expecting proper clearances and not common-english instructions. She needs to give them a simple HEADING and DIRECTION. Turn right heading 180 is all she needs to say. But instead she uses common english cardinal directions and on top of that screws it up, says it backward, and has to correct herself. No wonder they are second-guessing everything.
She then gives instructions to another aircraft, but you can hear EVA's crew trying to transmit in the background, possibly trying to confirm turn direction or heading, as we can only make out the words "left turn" in the snippet of their transmission, in between her transmissions to the other aircraft.
Again she tells EVA to "turn south now". No heading or direction.
Again she gives another aircraft instructions, and EVA tries to talk over them. When the other transmission is completed, you can hear EVAs crew saying "...right turn to southbound, maintain 7000". This transmission in fact does not sound like the same pilot who had been communicating previously (who would likely have been the non-flying pilot). This is possibly the other pilot, the one flying this leg of the trip, taking control of the situation and making a hard and fast decision to continue the right turn instead of trying to beg the controller for clarification on turn direction.
The controller responds "affirmative". The crew reads it back one more time to confirm.
The EVA 777 passes about half a mile south of Mt Wilson at around 6,300 feet and climbing. Mt Wilson is at 5,700 foot elevation with antennae that reach up to 6,600 ft.
The crew completes their turn to 180 (south), and informs the controller that they are heading 180 at 7,000. The flight continues on uneventfully.
I think it's pretty clear that this situation was started primarily by a controller giving a wrong turn direction (or not correcting a misunderstood clearance), but this situation went from "normal error" to "dangerous situation" because of the controller not giving standard ATC clearances to a foreign flight crew, rapidly changing the decisions that she was making, and choosing to use non-standard common language instructions that are not specific or clear enough for ATC communication. In addition, she made numerous mistakes of the tongue when communicating, exacerbating the situation.
Good readback awareness and clear standard ATC phraseology from the controller would have stopped this situation from evolving how it did.