Perhaps the main electrical issue is grounding of the secondary side. Since there is no neutral connection to the primary winding, when reversed it will have to be corner-grounded. With one phase grounded, the other two are then at 480V to ground instead of 277V. Additional measures might be needed to ensure code compliance.
Secondly the voltage selector taps will be in the secondary instead of the primary. If your incoming voltage is high, the transformer losses and heat dissipation might be increased because it won't be possible to increase the number of primary turns by changing the tap. However for voltages slightly above nominal this will be offset by the turns ratio compensation.
The compensation will however lower all the output tap voltages, by anything up to twice the percentage regulation. The turns ratio of a step-down transformer is deliberately made less than the nameplate voltage ratio. Off-load it does not quite step down enough, but on-load the copper loss drops the voltage to the correct value. Reverse-feed the transformer and the reduced turns ratio means that the off-load voltage will be low, and the full-load voltage lower.
Finally, and this might be the main issue, is that the manufacturer will probably disclaim all warranty and liability associated with the transformer, if it is not used in accordance with its specification. An inspector would probably side with them, if it were involved in a fire or incident.
Overall might work just fine but there are caveats with both performance and code compliance.