For certain subcategories of LCA groups, we have nice descriptions of the dual category under Pontryagin duality (all groups are implicitly assumed to be abelian):
finite groups $\leftrightarrow$ finite groups
discrete groups $\leftrightarrow$ compact groups
discrete torsion groups $\leftrightarrow$ profinite groups
discrete groups where each element is annhilated by some power of $p$ $\leftrightarrow$ pro $p$-groups
So I was wondering if we have a similar description of the Pontryagin dual of the category of abelian locally profinite groups, i.e. locally compact totally disconnected groups. Since locally profinite groups include discrete groups and profinite groups, the dual category will need to include discrete torsion groups and compact groups. Is there more we can say?