  A tremendous amount of research has been done in the last two decades on
$(s,t)$-core partitions when $s$ and $t$ are positive integers with no common
divisor. Here we change perspective slightly and explore properties of
$(s,t)$-core and $(\bar{s},\bar{t})$-core partitions for $s$ and $t$ with
nontrivial common divisor $g$.
  We begin by revisiting work by D. Aukerman, D. Kane and L. Sze on
$(s,t)$-core partitions for nontrivial $g$ before obtaining a generating
function for the number of $(\bar{s},\bar{t})$-core partitions of $n$ under the
same conditions. Our approach, using the $g$-core, $g$-quotient and
bar-analogues, allows for new results on $t$-cores and self-conjugate $t$-cores
that are {\it not} $g$-cores and $\bar{t}$-cores that are {\it not}
$\bar{g}$-cores, thus strengthening positivity results of K. Ono and A.
Granville, J. Baldwin et. al., and I. Kiming.
  We then detail a new bijection between self-conjugate $(s,t)$-core and
$(\bar{s},\bar{t})$-core partitions for $s$ and $t$ odd with odd, nontrivial
common divisor $g$. Here the core-quotient construction fits remarkably well
with certain lattice-path labelings due to B. Ford, H. Mai, and L. Sze and C.
Bessenrodt and J. Olsson. Along the way we give a new proof of a correspondence
of J. Yang between self-conjugate $t$-core and $\bar{t}$-core partitions when
$t$ is odd and positive.
  We end by noting $(s,t)$-core and $(\bar{s}, \bar{t})$-core partitions
inherit Ramanujan-type congruences from those of $g$-core and $\bar{g}$-core
partitions.
