A pervasive myth in the woodworking community is that you should finish both sides of wood, especially tabletops, to reduce the likelihood of warping. This myth has been carried to the extremes of finishing insides of chests-of-drawers, cabinets and drawers.
Especially for the undersides of tabletops and the insides of drawers and cabinets, it’s often a good idea to apply a finish for looks, feel and ease of cleaning. But the finish does almost nothing to prevent warps. All your experience tells you this.
For example, finishing undersides or insides of furniture made before the 1920s was very unusual. If tops from this early period have warped, as shown in the example, they have always cupped on the topside. This is the opposite of what you would expect if you think the warp could have been caused by leaving the underside unfinished to shrink more as conditions in buildings have become increasingly drier.
Plane sawn wood that hasn’t been dried adequately or has been moved to a drier climate can sometimes warp, adjusting to the drier atmosphere. But no finish, except possibly thickly applied epoxy resin, does any more than slow the process a little.
Deck boards laid randomly and finished on the top always cup, even though you would expect the boards to bow as the bottom sides dry out and shrink faster.
The true cause of warping, other than for wood that hasn’t been dried properly, is the topside or exposed side being wetted and dried out over and over causing “compression shrinkage.” The wood cells try to swell but can’t due to the wood’s thickness. So they compress to oval shapes and don’t resume their previous cylindrical shapes when they dry out.
The cells compress further and further every time they go through the swell/shrink cycle. The wood shrinks on that side, eventually cupping. If the cycle is allowed to continue long enough, splits develop. You can see this clearly on decks, which go through the wet/dry cycle countless times.
On tabletops that are frequently wiped with a damp cloth, it’s necessary to keep the finish in good shape to prevent moisture from getting to the wood. This is the way to prevent warping.