strong text>Hardly had we reached the station
This is an example of Adverb Fronting Inversion
Random Idea English at http://random-idea-english.blogspot.com/2014/09/exploring-inversion-and-fronting.html has a good explanation
Inversion and fronting
Inversion is often used in connection with fronting. Sometimes fronting involves inversion, often it doesn't. Sometimes that inversion is obligatory, sometimes it isn't.
• Fronting of a negative adverb, with obligatory inversion.
He had never seen such a wonderful sunset. (standard word order)
Never had he seen such a wonderful sunset. (fronted with inversion)
• Fronting of a prepositional phrase, with optional inversion
A large dog lay in front of the fireplace. (standard word order)
In front of the fireplace, lay a large dog. (fronted with inversion)
In front of the fireplace, a large dog was chewing a bone. (fronted, no inversion)
• Fronting of wh-clause - here inversion is not possible
I've no idea why she's late. (standard word order)
Why she's late, I've no idea. (fronted, no inversion)
The commonest uses are “so do I” and “neither do I”