As @chrisaycock mentioned, there's many permutations of parameters, especially when it comes to venue routing instructions, so it's hard to write an exhaustive list. But most of the time, the exceptions you're looking for will fall into 4 categories.
1. Intermarket sweep orders (ISOs)
This allows a destination trading center to execute against other orders on the book without checking protected quotes on other markets. For a neutral exposition of the details, you can check out SEC's FAQ on Reg NMS Rules 610 and 611. You can also see this enforcement action filing which describes its usage in a practical situation.
One important combination is an immediate-or-cancel ISO if you are crossing resting orders that are soon to be stale. And while the term "sweep" might imply that they are used for liquidity taking, a post only day ISO is an important combination if you are rebate-seeking.
2. Would-be locking orders
These are variations of order types whose price is expected to lock the market, and have predetermined behavior either at the entry time (when the order is posted) or at the time when contra-liquidity moves away so that the original limit price becomes permissible.
Often the follow-on repricing behavior can be emulated on client side, but a benefit of using the explicit order instructions is that you gain time priority by at least 1 round trip because the repricing or follow-on action is handled on exchange-side. (Note: This seems to upset a few people, but it's not different from using marketable limits, exchange-defined spread orders, or exchange-defined stop orders to take advantage of exchange-side logic.)
Such order instructions include Extended PNP Blind ALO on ARCA, non-routable Post Only on Cboe, price-to-comply orders on NQ/BX/PSX, among others. The names could use some marketing consulting from McKinsey, but the permutations are actually quite simple to visualize. You can probably build a flow chart with these questions:
- Should your incoming order be repriced upfront at entry or repriced later when the would-be locking price becomes permissible?
- How should resting hidden liquidity, on the opposite side of the book, at the locking price, affect the order's behavior before the repricing logic is handled? e.g. Can your incoming order display at the locking price?
- Can your incoming order receive price improvement if it would have crossed?
- Should your incoming order be canceled or kept on the book when it becomes permissible?
- If kept on the book, should your incoming order be hidden or not hidden, with corresponding matching priority, at the previously locking price when it becomes permissible?
3. Member-privileged orders
Variations of orders designed to systematically preference members of a particular status.
For example, the NYSE d-Quote specifically needs to be routed to an NYSE floor broker (not a "HFT firm") to be manually released into the closing auction, but allows a participant to send orders up until 15:59:50 (10s before the close). On the other hand, a conventional LOC or MOC has several restrictions: a 15:50:00 cutoff unless they are sending an offsetting order conditioned on significant imbalance. So this allows any buy-side firm to increase closing auction order imbalance or turnover the sign of imbalance through a designated floor broker.
4. Mass cancels
Because individual order cancel messages are not economical with serialization latency and messaging limits at a port/session level, there are often situations where a market maker would need to use mass or bulk cancels to pull multiple orders at once.
Though a little dated, I may also recommend this EDGA/EDGX order type guide for various pegging scenarios. I may also suggest this NASDAQ summary page of order types and modifiers.