Jupiter's magnetosphere extends "seven million kilometers in the Sun's direction and almost to the orbit of Saturn in the opposite direction" and Saturn's magnetosphere extends towards the Sun on average 22 Rs (Rs=60,330 km is the equatorial radius of Saturn). My question is: do these magnetospheres actually touch each other when the planets are on the same line?
A planet's magnetic field extends furthest in the direction away from the sun. This longest part of the magnetic field is called the magnetotail.
Jupiter's magnetic field is huge. It's tail extends up to 7,000 Jupiter radii, or about 490 million km per Wikipedia.
As closest pass, Jupiter's aphelion is about 816 million km from the sun and Saturn's perihelion, is about 1,352 million km from the sun, so they get as close as 546 million km. The distance Saturn's magnetosphere reaches forward would be a handful of Jupiter radii, but what the math suggests is that mostly Jupiter's tail doesn't reach that far, but it does reach most of the way to Saturn. There's some room for error based on how accurate the 7,000 Jupiter radii figure is. At the very least it nearly reaches Saturn. The fields might occasionally touch and exchange particles but probably rarely and not a lot of particles, given that it's the tail end of Jupiter's magnetic field.
As a sidebar, the Moon passes through the Earth's magnetotail.