I've discovered that the basic 4D rotatopes described by alkaline (cubinder, duocylinder, spherical cylinder) are by no means the only possible "wheels" in 4D. I found out that there are, in fact, an infinite number of 4D shapes that can roll like wheels.
Here's an example: the cylindrical double-cone:
Construction: take a 3D cylinder, and attach 2 cones (or half-cones, for the mathematically pedantic) to its circular ends. Now twist these two cones into the W axis until their apices meet. This produces a 3-member "ring". Now, take a triangle whose apex lies on the origin of the plane, and rotate it around the x-axis so that it forms a triangular torus. This torus can be fitted on the 3-member ring to make a closed 4D object: the cylindrical double-cone. (Provided, of course, that the size of the triangle matches with the length from the base to the apex of the cones.)
Properties: This object has 3 types of rotation. One is on its cylindrical volume which covers a straight line. The other 2 are rotations on the 2 nappes of the cones, which cover the area of 2 circles. It has 2 triangular faces which meet at one vertex, and are rounded off along their edges.
Here's another example: the tri-cylinder:
Construction: take a 3D cylinder and attach two other cylinders on either end. Twist these two cylinders into the W axis until their free circular faces meet. Now construct a triangular torus by rotating around the X axis a 2D triangle with apex facing the origin but displaced along the Y axis by the radius of the cylinders. The resulting torus can be fitted onto the ring of cylinders to form a closed 4D shape.
Properties: this object has 3 faces that it can roll on; the 3 directions it can roll on are 120 degrees apart. It has 2 triangular faces on opposite sides, which are rounded off at their edges.
An even more interesting shape is the tetra-cylinder.
Construction: start with a 3D cylinder, and attach two other identical cylinders to its ends. Rotate these two cylinders into the W axis until they are perpendicular to the first. Now attach a 4th cylinder between the free ends of these two. Now take a 2D diamond (a square rotated 45 degrees), displaced along the Y axis by the radius of the cylinders. Rotating this around the X axis gives a square torus which can be fitted onto the 4 cylinders to form a closed 4D object, the tetracylinder.
Properties: this object can roll in 2 perpendicular directions just like the duocylinder, but it has two round sides per direction (a total of 4 surfaces) that it can roll on. It has 2 square faces which are rounded off at the edges.
As you can probably tell by now, these last 2 objects are really just 4D prisms with cylindrical sides. We can attach N cylinders end-to-end, making a ring in 4D, and "cover" it up into a closed 4D shape by using a torus made by rotating an N-polygon. In this way, you can make prismic "wheels" that rotate in N different directions simultaneously. (Although they will only cover a 2D area.) If you take the limit of N to infinity, you will end up with a spherical prism which can cover a 3D volume by rolling. (This may be the same object as the spherical cylinder, but I'm not sure about that.)
There's also a bizarre object which I call a "bi-cone" for lack of a better name.
Construction: Start with a 3D cone, and stack smaller cones with linearly decreasing height onto it along the W axis until they vanish into a point, forming a 2nd apex an equal distance from the center of the circular base as the 1st apex.
Properties: this object is very strange, in that it has a sharp edge between its two apices, but projects into a 3D cone at 2 different angles. It can also project into 2 cones attached by their bases. Its circular base can actually rotate around the circumference; so it can act like a wheel if balanced correctly. Its sharp edge is perpendicular to this circumference, so it can roll and at the same time have its two apices point in the same direction, like 2 horns. :-)
Now, I'd be darned if I can come up with mathematical formulae for these objects... anyone know where to begin?