The principle is simple - it comes down to the realisation that, if one is
comfortable with Mills' Mess, one may make a right or left-handed pass starting
at any position (arms crossed or uncrossed) to their partner's right or left
hand. Note also that, from a rhythmic point of view, Mills' Mess is nothing
but a visual variation (crossing/uncrossing arms) of a normal cascade.
In essence, it is that a club thrown with the right hand will arrive in the
left hand after turning once (i.e. that it's siteswap notation is the same as
the cascade: 3).
From a rhythmic point of view, juggling in siteswaps fundamentally has no effect on the passing pattern. It's possible to continue juggling in the same pattern with one or both jugglers doing Mills' Mess while continuing to make the passes at the same time as they would otherwise. However, some supplementary remarks and advice might be useful.
This concerns passes (or rather the way they're done) made by the juggler on the other side of the pattern, whether he also does Mills' Mess or not.
Firstly, the passes should be made more to the centre than usual (especially in 4-count) so that they can be caught with the left hand, which is crossed to the right. Passing systematically tothe center avoids the problem of having to rack your brain over the exact place where the passes will be caught and is an ideal compromise for the person catching them.
In addition, normal passes are not easy to catch. Even when they are done well, passes spun in the opposite direction than normal are better adapted (since the person catching them doesn't have to twist their wrists so much). Below are several types of passes to illustrate my point.
(which should be avoided)
single reverse spin
3-count is a pattern that lends itself best to this exercise (Mills' Mess essentially follows a three count). Thus one throws the passes from the same position while alternating right and left hands. If the first pass is made with the right hand with crossed arms, right arm underneath (that's how I do it, thus sending the pass to my partner's right side), the second will be made with the left hand with the left arm beneath the right, then it repeats. It is always easier to make straight passes (if the right hand is on the left side, then the pass will go to the partner's right side), but if you want to alternate during a pattern between Mills' Mess and normal passing, you must find a compromise (in normal passing, a right-handed pass would go to the left side). One solution is to cold start with Mills' Mess and stay there (or if you want to switch to normal passing, go into Jim's 3-count).
In 4-count, the right hand will always be throwing passes, but never twice in a row from the same position. One will throw passes "arms crossed, right arm underneath," then "arms uncrossed" and so on. In 4-count, there is no problem going between Mills' Mess and regular passing and vice versa. The method that I use to go into Mills' Mess is to throw the first club from my right hand (i.e. the first right-handed self after receiving a pass) underneath (that is, under the left arm) and then the pattern just happens on its own.