Semi-cooperative game....thoughts on turn order

It depends a lot on how much of advantage it is to go first. If it is a big advantage, then you get some game balance by letting the person who is in last place go first. In that case, it is a disaster to let the person in first place go first, because then you have a runaway leader issue.

You said that in your game, it might be an advantage to go early or late, depending on other circumstances. There is a similar situation in the game http://boardgamegeek.com/boardgameexpansion/109140/fresco-bishops-favor (which I had to go look up on a friend's BGG list to get the name, so I figured I would go ahead and put the link in). In that game, the first phase in each round is choosing when to go to market (which really means choosing the turn order for the entire round). There are advantages and disadvantages each way, so it isn't a case where you would always want to go first, though sometimes you really want to do so. Anyway, the game works by making this selection in reverse order of score, so the person who is in last place gets first selection on the turn order chart. They have done a good job of balancing the whole spectrum of when you go (the earlier you go to market, the better the selection but the higher the prices -- if you don't have much money, you can still buy a lot of stuff if you are willing to take whatever is left after the other players have ravaged the place. Other times it might be important that you are at least second, but don't have to pay the extra cost of being first.)

Lots of worker placement games (Lords of Waterdeep comes to mind) have one of the activities for your workers gives you first action on the next turn. So there's a small opportunity cost to spending a worker, and the player has to weigh that cost against the benefit next turn. In all of those that I can think of, the play goes clockwise from there.

I've noticed that altering turn order can really help the players feel like they're in control of the game, and that players who enjoy deeper games really like games that mess with turn order. Casual gamers don't like you messing with what they already know. Bear this in mind when designing your game: most turn-altering schemes are a turn off to casual gamers.

However, I really like games with those mechanics. As such, allow me to summarize some that might spark some ideas:

Game of Thrones
When I heard of the Game of Thrones Board game, I rolled my eyes because I had never gotten into the series, and didn't honestly care for the theme. However, when I started playing with some friends, I realized that the game is a true gem.

As you play, if you don't have to order troops to attack, defend, etc, you can order them to collect power. For some, this is rather difficult, because you never have those free moments. However, power really becomes important because at random times during the game, you bid power in order to better position oneself along 3 different score tracks (using the term score loosely). The first is for "the throne" (I forget the actual name) which determines turn order. They also get to determine the results of all non-combat ties. The second determines who will win in the case of a tie in military combat, with a bonus to battle as well. The third lets you take back a preassigned move (not going into that), access duplicate moves, and see the future in a small way.

You all bid for power along these tracks by putting "Power" tokens into hand and simultaneously revealing them. The player who bid the most takes the top spot and the special token while the others are placed in order along those tracks. Turn order shifts as a result throughout the game, and can stay there for quite a while, so taking the advantage while you can is important.

Power Grid
I love this game. My wife does not. Oh well.

The players that have the most cities are moved to the lower positions (1 and 2) on the turn order track, and the players with the least cities move to the back (5,6). Having claim to cities is advantageous for a variety of reasons, and the game ends when a player reaches a certain number of cities. However, having more cities often hurts a winning player, because while you go in turn order for bidding on power plants (which sucks when you're first to bid), you go in REVERSE order for other parts, like buying up the resources (which are in a limited quantity, and the less there are, the more they cost) and building stations in new cities.

Smart players will intentionally stay a bit behind the pack in order to reap the benefits of being low in the turn order before they burst ahead and claim a lot of cities. It's all about when to make that power play.

Scum
This is one game where the turn order is DESIGNED to benefit the winning player, and yet people love to play it!

As you likely know, seating is determined by the order in which players remove all cards from their hand. Deciding when to start taking hands becomes very important, because you want to ensure that you get a chance to play your lower cards, but you also don't want to waste all of your big cards right away either or you'll most certainly end up last. Sitting closer to the king lets you reap the benefits of playing your low cards early (unless they're dealt an inhumanly good hand), but if someone a little lower down has the power to claim the trick and start anew, you get the short end of the stick and often end up with the ability to only play high cards.

This mechanic is great though if you want everyone to be ganging up on the winning player, because it just adds insult to injury to see the winning player then get the biggest advantage in turn order. If everyone gangs up on that player, then such advantages are mitigated.