Here's some examples of using accent patterns that run contrary to sticking patterns.
Most drummers that have been playing for any amount of time learn paradiddles and often accent the start of each set. RLRR LRLL. That helps us keep track of where we are in the sticking and it's also quite comfortable to play due to the double giving us time to set up the opposite hand for the accent.
There's no rule however, that we can't accent other parts of the sticking. George Lawrence Stone's follow-up to Stick Control (Accents and Rebounds) deals with this issue in depth.
We can also use the rhythms from the melodies to standard and Jazz tunes as accent patterns and apply it to different stickings.
For this, I'm using the melody to "Straight Ahead" by Kenny Dorham (from Una Mas), a great tune from a great recording. It goes like this:
YEAH!!! Anyway, here's me playing the melody in accents while playing singles off of each hand, (an A section each to save time) doubles starting with the RH in the bridge and doubles starting with the left for the last A.
That can be quite challenging in itself, but you gain lots of control over your accents that way, especially if the accent is on the first or second note of a double.
Here's the same melody in paradiddle sticking. The other great thing about this is it helps us commit sticking patterns to muscle memory while we concentrate on the melody....
Obviously, use any tune you like and also don't be afraid to work through the sticking/accent combos very slowly, and bar by bar if need be. I certainly have with many tunes!
Of course the stickings you apply to this can get as wacky as you want. Here's the melody with a LRR sticking which keeps circling over the bar line....
This gets very cool as you voice the sticking between 2 sound sources. Here is the same sticking between 2 toms...
Notice how the tonal pattern suggests one thing, while the accent pattern suggests something else!
Here's the same pattern between snare drum and hi-hat, which starts to lead us to Gadd/Garabaldi territory.....
Here's one more sticking (RLLL) and I've voiced this between rim of the bass drum and closed hi-hat with the left hand with bass drum on 2 & 4. It gives it a sort of "Kenny Dorham meets Kraftwerk" feel!
In closing, this can be quite challenging but you can gain tons of flexibility in your accents/stickings this way!