Here is a description of how Jupiter causes the orbit of Mars to precess. It includes enough detail to see which way the orbit precesses. I consider in particular the apsidal precession. The perihelion of a planetary orbit is where the planet is closest to the Sun. For Mars it is observed to move in the same direction as Mars does along its orbit but much more slowly.

First some useful observations as a lead up. Somewhere I heard the slogan, “A kick in the perigee raises the apogee, and conversely.”. (This is for satellites of Earth.) This is imprecise but can be made precise by clarifying that a kick is to add a Δv to the velocity of a satellite in the direction of its velocity. Δv is a vector change to the vector velocity of the satellite. The two vectors here are in the same direction. The kick adds energy and since there is no force, but the Earth’s gravity, on the satellite after the kick it is destined to return to the point of the kick. The only way the satellite can manifest its increased energy is a higher apogee.

Now we switch from Earth satellites to planets, and from apogee and perigee to aphelion and perihelion—same rules. Imagine a kick towards the Sun, Δv nearly perpendicular to the velocity, just before the planet reaches aphelion.


In this image the Sun is at the origin and the orbit of Mars, whose eccentricity is exaggerated, is initially the darker ellipse.
Mars proceeds about it orbit counter-clockwise.
Jupiter is much further to the left but it tugs on Mars more when Mars is at aphelion.

We choose the kick time and Δv so as to just reverse the small positive radial velocity that is decreasing as it approaches aphelion. That puts the planet on the fainter orbit with the same energy, and whose perihelion and aphelion are slightly rotated right as the planet orbits to the left. It is as if the planet bounced off of a hard spherical wall centered on the Sun with radius slightly less than aphelion. The bounce was elastic and no energy was lost. All the classic orbital elements are the same except for perihelion.

Consider the effect of Jupiter on the orbit of Mars. Jupiter’s mass is about 1/1000 that of the Sun. Where our experiment above kicks Mars towards the Sun, Jupiter delivers a slow kick to Mars away from the Sun. This rotates the orbit of Mars to the left. Our image above can serve this scenario as well if the initial orbit is the upper faint ellipse. As it first gets to the wall we make a small hole it the wall to let it thru. It comes back and hits wall and gets an outward kick bouncing but not losing energy or tangential momentum. It finds itself then on the darker ellipse; the orbit has precessed to the left which is the direction of Mars in its orbit, and the observed direction of its precession.

This establishes the sign of the precession of Mars. See the time derivative of ϖ for Mars in the 2nd table. More quantitative detail could establish an order of magnitude.

We consider the effects of radial kicks. The radial velocity is 0 at aphelion which is why the planet is near aphelion.
In the image the first ellipse is r=1/(1+0.7cos(θ)). The 2nd is r=1/(1+0.7cos(θ−0.2)). The circle is r=3.29 .