I was curious about this myself, and since the earth is not transparent the only way I could think of to examine what is underneath the solar system is to go to the other side and see for myself.
I did this, and what I saw should have been predictable in the main but surprised me anyway. Some aspects were just a little unsettling and one part was downright shocking.
Predictable… In that below us are more stars and stuff that is similar in appearance to what is above us (except that they are different stars of course, and they are in different arrangements compared to on top. For example the North Star cannot be seen, which is awkward for those of us who routinely use it. Why there is not a South Star is beyond me. There are a lot of them, and it would have been just as easy to make one be the South Star. Most of the familiar arrangements of stars cannot be seen from there, but there are some interesting ones that you can see that are not visible from on top because (as you point out) the earth is opaque.
There are quite a few permanent colonies of people living underneath, and the residents of those colonies seem to be obsessed with the underside of the solar system, as their ceremonial cloth banners carry a depiction of it.
The thing that really shocked me is what they are hiding down there. I left the colony that was hosting me, and traveled far from it so that the light they created during the night would not interfere with my view of the underside. When I stepped out of my car, I was shocked to see that there is a GALAXY! I mean IT IS SITTING RIGHT THERE. I’m not talking about close like “yeah, you can see it pretty well…” I mean that I kinda think we are actually sort of in it.
In fact, after I gave it some thought, I realized that quite possibly the dense streak of stars you can see from the topside is actually the other side of the galaxy— the stuff that is farther from the center than we are already!
If you don’t believe me, I recommend you go look for yourself. I mean… we’re practically touching it.