This is something I'm learning to do: I'm practicing, and I hope that with a lot more practice I might become good at it. It's fun.
The floor-patterns (and partner-interactions that follow) and variety of steps in the mutanze of the passo e mezo are often delighfully varied, and really fun. The rhythm - or the most obvious rhythm - can get a bit boring.
Galliards and canaries have a syncopated rhythm built into their music that keeps interest even without any further effort from the dancer (and which often inspires further variation from the dancer); passamezzi have no such device. The music, as it appears in our dance-manuals, is straightforward, unvarying duple*: it's a lovely piece, but the beauty doesn't come from an exciting rhythm.
A passamezzo mutanza, performed always exactly on the beat, with the steps arranged in the most natural way to fit the music, can become a little rhythmically boring. Beautiful, exciting variety in the steps can be reduced to "jump, jump, jump, jump, jump, jump, JUMP ... [repeat]".
One solution to this - especially in the denser, more complex mutanze (where you typically do something on every available beat) - is to deliberately move some steps just off the beat: linger over some, hesitate before performing others, rush a little here, wait a little there, and perform some passages exactly on the beat for variety. For a given mutanza, there are a lot of options that will produce an elegant, witty or pleasing result without overdoing it to the point of tastelessness.
In practice, I find that I learn a mutanza first - memorise it, get it into my feet - by dancing it very "straight". I memorise a patter so I can call it for myself, and dance it in a straightforward way, without much concern at first for elegance or variety. Once I can execute the steps easily and with barely any conscious thought, I start to play with the rhythm, and I try many different options. Sometimes I settle on one favourite, sometimes I try something new each time.
Here's the man's first mutanza in Caroso's Paso e mezo from Il Ballarino, and my current-favourite way to adapt the rhythm:
I think the trick is to introduce enough variation to keep interest and show skill, without so much that it appears random, or seems that you can't find the beat. Some well-chosen passages precisely on the beat demonstrate that any deviation is intentional. Some repetition (e.g. hesitating on the first part of the riverenza presta in several places in the mutanza above) gives a sense of unity. Returning to the beat for the finale finishes things off cleanly, and can be dramatic.
* There are instrumental passamezzi from the period that are rhythmically exciting: e.g. Diego Ortiz' variations on the passamezzo antico - he uses many suspensions, melodic phrases which start on off-beats and run across bass-line phrases, and one entire section which changes abruptly from duple to triple time.