For each axis, compute whether the intervals spanned by the two boxes are...
0: separated (minA > maxB or minB > maxA)
1: adjacent (minA == maxB or minB == maxA)
2: overlapped (minA < maxB & maxA > minB or minB < maxA & maxB > minB)
Return the least result you found on any axis.
(ie. The boxes are "overlapping" only if they overlap on all axes. Being "adjacent" on any axis trumps overlapped, and "separated" trumps all)