The answers so far are really interesting. Most on the right are basically responding with "Secession good, it's moral, it's right, it's freedom!"
OK, cool story. I won't argue, I'll accept your premise at face value. But it still doesn't answer the question. This is about the alleged liberal/conservative alignment of the two big parties, using positions on secession as an indicator. So all that really matter is whether supporting a unilateral power to secede is a liberal position, or a conservative position. If it's a liberal position, why do modern day conservatives support it? If it's a conservative position, why were the allegedly liberal Democrats of the mid 19th century supporting it?
I'm not seeing the concept of secession as a "liberal" or "conservative" position per se. In the instant case of the Civil War, it was used as an avenue to
sustain the conservative position, that of the preservation of Slavery and the plantation economy. Perhaps in that instance the dynamic could be called radical conservatism, as secession is a radical step.
It would presumably be plausible if, say, that conservative Slavery system had won the day in 1860, amassed all the political power and was steering the whole country and its territories in that direction, that the Liberal North could have decided "this won't work" and seceded itself, to protect its own Liberal values. So the act of secession, by itself, seems like a neutral.