Do all the centroid polygons's overlay on top of the corresponding polygons with actual boundary? Then Spatial Join tool should resolve your issue.
Keep the Actual Boundary Polygon layer as the target features, and the Centroid polygon layer as the Join features.
Join Operation: One-to-One
Match Option: "CONTAINS" or "COMPLETELY CONTAINS"
You may then delete the field(s) that you don't need in the output.
Think Location