As far as we know Earth and every other planet in the solar system rotate around their axes. Whether it be Uranus which rotates on its side or Mercury which rotates perfectly straight, all planets spin the same way. However, what if the planets rotated up and down? To explain this imagine if Canada slowly moved northward towards the north pole and once crossed started going south into the southern hemisphere and once it reached the south pole it started going north, Tl.dr the planet rotates longitudinally instead of latitudally. Is it possible?
No. The way we define latitude and longitude is based on the planet's axis of rotation. You can certainly (AFAIK) have a planet with a 90° axial inclination, or (probably) a planet that is identical to Earth except with all the land masses rotated 90°, but unless you completely change the definition of "pole", the poles will, by definition, be stationary w.r.t. the planet's rotation.
I'm going to assume that the question you're asking is 'Can the Earth additionally rotate up-and-down?', because, Matthew's answer has pointed out, we define 'up and down' based on the planet's already spinning axis. The only way this question makes sense is that you're asking whether the Earth can additionally spin up-and-down. And the answer to that is a clear 'No'.
This goes more into it, the short of it is that multiple vectors in a closed system will eventually become a single vector and basically the Earth will spin diagonally from our current perspective.
As you say, Uranus does. All it would mean is that the axial tilt would be 90 degrees. This is unlikely in the course of normal planetary formation, so (as with Uranus) it would probably take a series of impacts early in the planet's history to point its axis in the appropriate location.
Now, if we're talking about Earth, things would obviously look a lot different. The north and south poles would be two points on what is currently the equator, and the ice caps would be there as well. Also, given that this would've been the case for most of the planet's history, the equatorial bulge would have moved as well.
Yes, for a given definition of 'North Pole'.
You see, there are three different 'North Pole's
- Geographic North Pole, which is defined as 'the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface' (ignoring precession).
- Magnetic North Pole, defined as 'the point on the surface of Earth's Northern Hemisphere at which the planet's magnetic field points vertically downwards'. This is what your compass points at.
- Geomagnetic North Pole: the Magnetic North Pole and Magnetic South Pole are not actually directly opposite each other. This is the closest approximation to 'but, what if they actually were?'
As should (hopefully) be pretty obvious, the first definition is completely incompatible with your question: as soon as you move the axis of rotation, the Geographic North Pole moves with it. Canada cannot, by definition, rotate through the Geographic North Pole, and on down to the Geographic South Pole.
However, the Magnetic and Geomagnetic North Poles move. They are also largely independent of the Axis of Rotation. In the 1850s / 1860s, they actually got all the way down to King William Island in Canada - that's a quarter of the way to the equator! And, sometimes, Magnetic North is in the Geographic South.
So, you could (theoretically) have a situation where Magnetic North was pretty much on the equator. Countries could then rotate 'North' (Magnetically) past the Pole, and continue 'South'. However, they are still rotating around the planet's axis: For Magnetic North to be roughly where Geographic North currently is, and Canada to rotate through it, you would need the Geographic North Pole (or the Geographic South Pole) - and the Earth's Axis - to be somewhere around Cape Town
It should be noted, of course, that this would make Canada nice and warm, while Hawaii would be covered in ice.
Unlikely - but Yes
So taking this question to be "Could a planet rotate around an axis in the elliptic and tangential to its orbit?" The answer is:
"Sure - but it takes very specific setup."
So the planets all rotate because the swirling cloud of gas that they formed from rotated. Momentum is conserved, and there is nothing that opposes this motion.
Since the gas cloud started as a disk, the planets all start rotating in the same direction - on an axis that intersects the elliptic at a right angle.
You get something like Uranus because some large impact or series of interactions basically tilted the planet over 90 degrees. Now the axis is along the plane of the elliptic, and points towards the sun. The planet appears on its side.
To get a planet that rotates "Up and down" as OP described, you would need a second interaction or series of interactions that would change the axis of rotation be another 90 degrees in a different plane.
So you would now have to rotate the axis from pointing at the sun, to tangential to the sun.
Obviously these two interactions are very specific and very energetic, and would therefore be extraordinarily unlikely. But in a large enough universe of cases it could happen.