We introduce shot-profile migration data reconstruction (SPDR). SPDR constructs a least-squares migrated shot gather using shot-profile migration and demigration operators. Both operators are constructed with a constant migration velocity model for efficiency and so that SPDR requires minimal information about the underlying geology. Applying the demigration operator to the least-squares migrated shot gather gives the reconstructed data gather. SPDR can reconstruct a shot gather from observed data that are spatially aliased. Given a constraint on the geological dips in an approximate model of the earth's reflector, signal and aliased energy that interfere in the common shot data gather are disjoint in the migrated shot gather. In the least-squares migration algorithm, we construct weights to take advantage of this separation, suppressing the aliased energy while retaining the signal, and allowing SPDR to reconstruct a shot gather from aliased data. SPDR is illustrated with synthetic data examples and one real data example.