During a fight at a TV studio, the camera operator switches the camera to "Fight Scene", and the camera tilts to the side in a 'Dutch tilt', used throughout the 1966 TV series and film to depict the 'crooked' hideout.
One of the fight scene onomatopoea that comes up on screen in the first fight is "SPRANG!" This could be a reference to comic book artist Dick Sprang who helped redesign the Batman Ile, created the Riddler, & was one of the top artists on Batman comics in the Golden Age of Comics.
The trivia items below may give away important plot points.
The villains that Robin and Catwoman break out of prison to help defeat Batman when he goes bad are: Mr. Freeze, The Mad Hatter, False-Face, King Tut, Bookworm, The Archer, The Minstrel, Ma Parker, Egghead, Shame, The Sandman, The Siren, Clock King and Louie the Lilac.
When "Evil Batman" is replacing much of Gotham with his duplicates, he goes into a court & quotes the case of "Semple vs Dozier." This is a reference to show creator/narrator William Dozier & head script writer Lorenzo Sample, Jr for the original 60s series.
During the Gotham Palace fight scene, both of the ladies dancing alongside Batman have a hairstyle that actually was created in the 60s and based on Batman's cowl; this may be the same style referenced in a 1966 issue of Life magazine.
When Batman is under the influence of the Batnip on the space station, and he acts more like the modern Nolan and Miller versions of Batman, the "normal" 60's fight-sound words like "Splang!" and "Zot!" are replaced by the words "Rip", "Torn", "Gore", "Fracture", "Bludgeon", and "Pulverize".
After Batman goes bad from the Batnip and he tells the Commissioner and Chief O'Hara that he's going to keep the duplicating gun in the Batcave, after the Commissioner praises him with the standard overblown compliments they turn and find Batman gone. This is a textbook Batman move in nearly every other version of Batman, where he takes advantage of a distraction to mysteriously vanish, but the 60's TV version never did it. However here, all three, Robin included, remark that it's not like Batman at all.