When the landing party is in the middle of the cross-universe transfer, the USS/ISS Enterprise is seen in orbit going in opposite directions (USS Enterprise going counterclockwise [when viewed from above]; ISS Enterprise going clockwise). However, through the rest of this episode, the ISS Enterprise is going counterclockwise.
In beginning scene when Kirk and the others first beam back up to the Enterprise, Kirk signals the ship to engage, then puts his communicator away and puts his hands down by his side. As they first appear in the transporter pad aboard the ship, he has his communicator still in his hand, held up to his mouth. During the original series, movement during transport was not possible.
The goof items below may give away important plot points.
As Mirror Spock prepares to transport the landing party back to its rightful universe, he tells Capt. Kirk, "You have two minutes and 10 seconds." As Kirk tries to convince Mirror Spock that both he and his empire are "illogical," Mirror Spock tells him, "You have one minute and 23 seconds," meaning that 47 seconds have passed. But only about 30 seconds have actually elapsed.
When Chekov enters the bridge turbolift just before Kirk (just before his assassination attempt), his shadow can be seen from the standard bridge camera view, indicating a bright light source to the left of the turbolift car and a missing wall.