Sounds like Symmetric, except that it saves some space by not storing upper entries (as our implementation currently does) nor diagonal entries (which are assumed to be zero). I’d start by making Symmetric more efficient as regards memory layout in Julia Base. Not storing diagonal elements sounds like a secondary issue since for large arrays the cost will be relatively low.
LAPACK supports a packed storage format for symmetric matrices, so you could still get high performance for common linear algebra operations (e.g. matrix * vector product). (These optimized operations can be added progressively.)