It's the devil you know, as a long-term employee, versus the one you only know as much as you were able to glean from interviews plus other background research you may have done.
For this reason I've always taken the benchmark of a minimum 10% salary package increase or it is not worth the risk of changing company. Factor in the loss of holiday and other perks in working out the minimum acceptable %age to you.
I would say it is really unusual to try to head-hunt someone, mention a salary at first interview then reduce it as an offer without explanation.
You can (and should) broach it with the company - by phone - and prepare yourself well for the conversation beforehand. Along the lines of your enthusiasm for the role and the company but surprised and disappointed with the offer. It would be perfectly reasonable to ask what band or grade you would be fitting into, and what the top and bottom of that grade is (but every company has such structure, but large ones invariably do). You could also reasonably ask what the top and bottom of the salary band one below and one above is.
I wouldn't make a counter-offer - just say that you like the role but wouldn't feel able to accept the offer for the salary on offer. The ball is then in their court.
Bear in mind that if they head-hunted you, there probably isn't a second candidate in the wings - they have to start the recruitment process again.