You could do this, but considering that you would be skipping the first half of the show, you would definitely be missing out on more than a few key events. A lot of the initial relationship build-up occurs in the first half, so the later events might have less of an impact if you skip the parts before it.
Now if you were to watch the second half before buying the first half, would you go back and watch it again all the way through? I sort of wonder how it will change your perception of the first half when you already know what's going to happen later on.
Alternatively, if you're in the US you can watch the first half of the series streaming on Hulu before jumping into the second half of the series on DVD, if that works for you. I do understand everyone's desire to own the Premium version of Toradora, since it is a really nice release.