Does the hex spine (S6/T5) count as adjacent or not?
Although in ASL units can move along hexsides, they can only do so when bypassing a woods/building obstacle [A4.3, D2.3]. Movement along the S6/T5 hexside is not possible because neither hex contains a woods/building. If not using bypass, units move from hex center to hex center like other wargames (although LOS can sometimes be traced to points other than the hex center [A6.12]).
Even when it is using bypass, a unit is always in one of the two hexes. If the particular move is unambiguous (e.g. in your picture moving along the T4/U5 hexside) the player usually won't declare the hex. Where the hex is ambiguous the player must state which one is being bypassed. The choice of hex being bypassed can affect things like range and point blank fire. A unit bypassing the R5/S5 hexside in hex R5 would be at range one from R6 and so subject to the doubling for point blank fire. A unit bypassing that same R5/S5 hexside in S5 would be at range two, and so not subject to doubling for PBF.
JR