  We explain a "curious symmetry" for maximal $(s-1,s+1)$-core partitions first
observed by T. Amdeberhan and E. Leven. Specifically, using the $s$-abacus, we
show such partitions have empty $s$-core and that their $s$-quotient is
comprised of 2-cores. This imposes strong conditions on the partition
structure, and implies both the Amdeberhan-Leven result and additional
symmetry. We also find a more general family that exhibits these symmetries.
