Can you shuffle your discard pile?
edit: The OP was not clear whether it meant "shuffle the discard pile and put it back in the draw pile" or just "shuffle the discard pile". Now that's clear, I'm adding the following:

True, as the rules stand right now, the order in the discard pile does not usually matter (sometimes it does, see below). Theoretically, you could shuffle the discard pile at any time and just keep it away from the draw pile. The rules do not let you still do it for the following reasons:


The status of your discard pile might still change before the "proper" reshuffle, thus you might still have to reshuffle. Consider the following example:


Draw pile: 0 cards
Cards in hand: Market, 4 others


You are going to play the Market, which lets you draw a card, so why not shuffle the discard pile while waiting for your next turn? Well, you might receive a Curse until then, so you'll still have to reshuffle with the Curse in it.
Reserved for future "discard pile" rules. Currently, there are no cards that do something, say, with the top card of the discard pile, so the order in it does not matter. Despite that, the rules still don't explicitly let you shuffle arbitrarily, because DXV likes to have as much room as possible for future cards. For example, one day he could design a card that does something with the top card of the discard pile.
For the sake of completeness, there are actually cards that care about the order of the discard pile: Watchtower. When you gain a card, you may reveal the Watchtower to trash that card or top-deck it. If you do, then the Watchtower expects to find the card on top of the discard pile. Consider the following:


You have a Watchtower in hand.
You gain an Inn.
Now, there are two things that happen at the same time: you either trigger the Inn or the Watchtower first.


Suppose you resolve the Inn first.
You choose as many action cards as you want from your discard pile and you shuffle them in your deck, but not the Inn.
If you leave the Inn somewhere inside your discard pile, but not on top, then you may not reveal the Watchtower to topdeck/trash it, as the Watchtower has lost track of the Inn.
Nevertheless, if you leave the Inn on top, then you may reveal the Watchtower (source).






According to the base-game rules, you only shuffle the discard pile when you need to do something with the draw pile (e.g., draw cards, reveal cards, etc) and there are not enough cards to do so. You cannot arbitrarily shuffle the discard pile whenever you want to, for a couple of reasons, including:


An opponent likely holds a Witch (or any curser) in hand. Your draw pile has been depleted and you hold your last 5 cards in hand. You would like to reshuffle immediately, so that the Curse will "miss" the reshuffle and will not see it soon.
You are ready to buy your first Province and you're close to reshuffle. You'd like to reshuffle immediately, so that you won't see "soon" the Province.
You have just reshuffled and played a key card (e.g., Witch). You discard the Witch, but you have over 30 cards in your deck. You'd like to immediately reshuffle your discard pile so that you'd see the Witch again.
Any cards that you gain go immediately to your discard pile (unless stated otherwise). Thus, if you shuffle the discard pile, these cards will enter the draw pile, whether you'd like it or not.
There are cards that do something with the discard pile, like Counting House, Hermit, Inn, Herald, etc. Arbitrarily shuffling the discard pile (or not) directly affects these cards.


Thus, there is a general, fair rule, that makes sense as well: you shuffle only when there are not enough cards to do something.