Unfortunately, I feel the same way

That can't be the case. Send it back if it is still under warranty and otherwise buy a new one.

The display rotates, it is always legible. When it is no longer connected to the power supply, the display switches off and is reactivated with every touch (including shaking the table). The battery lasts a very long time (3 months), ideal as a travelling alarm clock.

Battery full. Completely clear

My Lexon Flip Premium works correctly whether it is turned "ON" or "OFF". Make sure it's fully charged; otherwise it's a fault.