In single cell RNAseq do we also list down regulated genes as key signature genes of a particular cell type?
All analysis serves to your project goal. It all depends on what information you are trying to extract, and whether it is required for supporting your hypothesis.
You certainly can, people just tend to show those upregulated in a given cell-type/cluster, as it's arguably easier to interpret.
I agree. Also keep in kind that a gene being downregulated means it is upregulated a celltype you compared it with. Presentation of genes being up- or downregulated should imho be based on the message you want to send. If it makes more sense to report downregulated genes I see no reason not to do it. There is definitely no rule that forbids it. Whatever floats your boat :)