Is there a backside of a black hole? From all that I can gather, a black hole is thought to be spherical. In this case, would it be theoretically possible to travel around (not inside) the black hole in order to see the backside? If so, would the backside look the same as the front side (i.e. a black hole)? Everything I read about black holes tends to be about what's inside them, which is not what I'm asking.
To make this even more clear, let me explain with an example of earth. Let's say we're on a spaceship traveling around earth, we look down and see that we're over the United States. We decide that we'd like to travel around the earth to see the backside (or opposite side from our current location), which would be somewhere in the Indian Ocean—between Australia and Africa. We have many ways to travel there; we could go over the north pole and down, down by the south pole and then up, or go east/west to get to the other side, etc. Now, would this be theoretically possible to do with a black hole? Yes, I understand that if we get too close to the black hole we'll be sucked in, so let's just say we stay far away enough to not have that happen.
 A: A "normal" Schwarzschild black hole is spherically symmetric. If you had a ship that could travel around the black hole, and provided the ship has sufficient propulsion so that it wont touch or pass the event horizon, then yes. You could see all "sides" (there are no sides since once again it is spherically symmetric, and no area or surface is different to another).
Now a rotating black hole will have a rotation axis, and so we lose the spherical symmetry we had with a normal (Schwarzschild) black hole. But we will have axial symmetry (which means rotational symmetry around an axis of rotation), similar to earth. So you would have a "north pole" and "south pole" as you put it, so flying from one pole to the other might prove to be interesting.
General relativity predicts that rotating black holes exhibit something called frame dragging. Frame dragging means that objects moving close to the black hole will be forced to rotate with the black hole, because the actual spacetime around the black hole is being "dragged". So you may want to keep your distance a little (lot) more greater than the event horizon.
Also, rotating - Kerr black holes - have a region, called the ergosphere, where the spacetime around the ergosphere is "dragged along" by the black hole, but this time even faster than the (local) speed of light. You would want to be even more cautious about that or you might find yourself on a perpetual "merry-go-round"!
Black holes also have regions called the photon sphere which is light from the accretion disk. Remember the first image of a black hole in M87? The reddish region. The photon sphere is about $2.5$ times further than the event horizon. You also need to take this into account, and the other points above, since your assumption that it would be safe to fly around a black hole provided you don't cross the event horizon, needs urgent reconsideration!
If you unfortunately happened to fly inside the photon sphere, if you were looking directly ahead, you would see the back of your own head, which would probably cause problems for you whilst attempting to navigate! :) (The light coming from the back of your head would  orbit the black hole, and then come back to your eyes).
Also, watch out for Hawking radiation. Not that it is energetic, and would cause any problems (as far as I am aware, it is less energetic than the cosmic microwave background radiation), and given that it is emitted in all directions isotropically, but it would be interesting if you could give us confirmation of its existence. :)
Does a black hole have a backside? Probably not, and depending on the type of black hole, they do have parts/regions that are different (except for a Schwarzschild BH) to others, and some more interesting than others.
A: As you say, a black hole is spherical and doesn't really have a "front" or "back" side any more than The Sun (for example) has a front or back side.  The black hole itself looks the same from all directions - a black hole with only Hawking radiation escaping.
Around the black hole, however, there will almost certainly be an accretion disc of matter spiraling down into towards the singularity which will be rapidly giving off all sorts of energy.  I would suggest watching this video - https://www.youtube.com/watch?v=zUyH3XhpLTo - to further understand what it looks like from an outside observer.
