There are several reasons. See here...
LSR nut view from tuner side
First, you have rubber dampers on the tuner side. If you flip it around, you'll damp the strings on the playing side.
Second, the slots the strings feed through are spaced for right-handed stringing. That is, low E on top, high E on the bottom.
Third, the strings rest in the "V" created between the bearings, which aren't centered on the block. You could compensate for this when you cut the slot for the nut, but it'd be risky business.
Finally, the mounting screws are also offset to the tuner side. You could compensate for this on a new install, so it's not a deal-breaker. Just something else making it non-symmetrical.
You may want to look at Graphtech nuts. They're not the ultimate part the LSRs are, but they're pretty slippery. If they're well-cut, string hangups aren't an issue. Also, don't install string trees. They're only necessary if you have a poorly made nut.