Here, Prateek raises the issue of the border states. This is a good point, but I don't think it does much to inform the "cause" of the war. It's fairly widely accepted that the Union itself was not especially concerned with abolishing slavery. In other words, the motivations weren't necessarily symmetric.
A better point to raise, I think, is the Confederate discussions of emancipation during the war. These discussions didn't come to anything substantial, but it does raise important questions about how important a war aim slavery really could have been to a society discussing abandoning it. I've heard discussions about how this doesn't really prove anything because even the advocates still envisioned and were trying to defend a post-war South governed by white privilege. I'm not particularly impressed with that answer. After all, the entire country - North and South - envisioned a society governed by white privilege. The fact that Southerners felt that way too hardly tells you anything about war aims (few Northerners, abolitionist or otherwise, didn't envision their own society as being governed by white privilege).
I think what the Confederate emancipation plans point to is simply the heterogeneity of Confederate opinion. Some considered the institution of slavery considerably more important to the South than others.
Oh... and worth noting.
Where were the proponents of emancipation from? Cleburne was from Arkansas, Lee was from Virginia.
Where were the opponents of emancipation from? Davis was from Mississippi and Walker was from Georgia.
If you think this is a coincidence, I think you are being foolish.