Osgood continues to be a living homage to previous incarnations of the Doctor by wearing a shirt with question marks on the collar, also worn by the 4th, 5th, and 6th Doctor.
There is a real town in New Mexico called "Truth or Consequences." The town's name was changed from "Hot Springs" on 31 March 1950 after Ralph Edwards, the host of the popular game show "Truth or Consequences," called for any town in America to do so - in celebration of the show's ten year anniversary.
The Doctor remarks that he once "snogged a Zygon". The Tenth Doctor kissed a Zygon impersonating Queen Elizabeth in The Day of the Doctor (2013).
Kate Stewart being unsure over which decade the Zygons first attempted to invade in, the '70s or the '80s, is in reference to the ambiguity over whether or not the original UNIT stories took place in the present or the near-future of their mid-70's air dates. The invasion that she specifically mentions was shown in the Tom Baker serial Terror of the Zygons: Part One (1975).