Is this statistical group summation unambiguously reversible?
The answer is no . I found it helpful to think of the information in the form of a picture: mark each pair $(a, b)$ as a point on a grid, then in each column and row write both the count of the entries in that row, and the &quot;sum&quot; of the entries (the sum of the 'other' coordinate). Your example may be pictured as follows:

I'm writing the count of the dots in each row/column as the number closest the the column, and the &quot;sum&quot; as the number further away. The original data is the set of dots, the new data are Y and Z (understood to not just be sets of points, but to be positional: for example we remember that in the $Y$ data we have $(2, 3)$ in column $1$, $(1, 3)$ in column 2, and so on).
Here are two configurations of points with equal $Y$ and $Z$ data:

Hence the operation is not reversible. Per-column ambiguity arises when there are multiple integer partitions with the prescribed number of parts: eg if a column counts to 2 and sums to 5, its possible entries are 1 + 4 or 2 + 3.