Is there some way to look at the front of my character? And I don't mean looking at the image in the character menu.
Left click and pan your mouse.
Only works when you're standing still, unfortunately (when you start to move, the camera resets to behind you).
You can also bring up the preview window for gear, and spin your character (or companions) around in there.
Ctrl+Left click on any bit of clothing (doesn't matter if you're qualified to wear it or not) and the preview window pops up. Ctrl+left click on that clothing again and it's removed and the preview just shows what you're actually wearing right now. Click the arrows at the bottom to spin your character around, or click the companion tab and use the arrows at the top to swap through your companions.
This also works for gear that's still on vendors, being offered in the mission completion window, or the icon of gear in the crafting window. Just be careful on vendors - Ctrl+left click on gear in vendors and you preview it, Ctrl+right click on gear on vendors and you buy it (pretty sure that one of the loading screens wrongly tells you that ctrl+right click is used to preview, this is how I ended up accidentally spending 8K on a cool looking Dark Side III Sith robe, that my purely light-side Trooper can't wear as I was walking to my trainer with just enough money to pay for speeder training :( ).