Partition theory abounds with bijections between different types of partitions. One of the most famous partition bijections maps each self-conjugate partition of a positive integer $n$ to a partition of $n$ into distinct odd parts, and vice versa. Here we prove new necessary and sufficient conditions for a self-conjugate partition to be $t$-core, in terms of only the parts of the corresponding partition into distinct odd parts, by proving a new hook length formula. Corollaries of these results include new applications of $t$-core self-conjugate partitions to subsets of the natural numbers, due to the recent investigation of a new partition statistic called the supernorm by the first author, Just, and Schneider, as well as many results on $t$-cores by Bringmann, Kane, Males, Ono, Raji, and others. We provide several examples of these applications, one of which gives a new formula for certain families of Hurwitz class numbers.