Are there known centaurs which orbit between Jupiter and Saturn without crossing their orbits?
The JPL Small Body Database lists 5 asteroids and 5 comets with q > 4.56 au and Q &lt; 9.04 au.
Two of those asteroid orbits have large uncertainties due to few observations and may not actually meet your criteria. These are fairly well determined:


Asteroid 494219 has a highly inclined 14-year orbit which occasionally brings it within 2 au of Jupiter. It was discovered in 2012.
Asteroid 2000 GM137 has a 22-year orbit which occasionally passes within 2 au of Saturn. It was discovered in 2000, unobserved for a few years, and recovered in 2014.
Comet 29P/Schwassmann-Wachmann 1 was discovered in 1927, has a nearly circular 15-year orbit, and is known for frequent outbursts.
Comet 39P/Oterma was discovered in 1943 on an 8-year orbit. A 1963 encounter with Jupiter switched it to its present 20-year orbit, where it became dormant and was lost until 2001. Numerical simulations indicate a prior Jupiter encounter in 1937; it was likely also a Centaur before that.