In opening scene, when the landing party first beams 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 in mid-beam, you can see he still has his communicator in his hand, held up to his mouth, and when he finally materializes, his hand is back down by his side. (This is clearly a special effects error, as, in the original series, movement during transport beaming was not possible.)
In the turbolift, Chekov is missing the sash he's wearing both before and after.
When the crew members are first transported to the alternate universe, the scene shows the Enterprise flipping back and forth between orbiting left-to-right and orbiting right-to-left, to symbolize that the crew is now in an alternate universe. However, the outside of the alternate Enterprise has the traditional Starfleet Logo, rather than the alternate-universe's Earth-and-Sword I.S.S. logo that is on all the interior walls and the uniforms.
In the first assassination attempt by Mirror Chekov, Captain Kirk is hit in the face by someone wearing a blue shirt. When the scene zooms out, that person isn't there anymore.
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.
After the fight, Mirror Spock is unconscious on the table. Mirror Sulu shows up and does not have the big ugly scar on the right side of his face.
Poorly-placed camera angles reveal the fight sequence choreography between Kirk and the "mirror" Spock. On more than one occasion, very noticeable gaps are visible as they fight.
When evil Kirk is yelling at Spock as he walks away, Kirk moves his head through the plane of the door, straight through the supposed force field.
The communicator scenes between Kirk and Scott following Spock informing Kirk of his orders to kill him. Scott is obviously reading his lines from cue cards. In all his deliveries, he is looking at a fixed point to his right.
During the fight in Sick Bay, Spock's (Leonard Nimoy's) stunt double is rather obvious. His hair is not styled in typical Vulcan fashion, but is fuller and wavy.
When the landing party is beamed to the alternate Enterprise, they are wearing alternate universe uniforms. The regular uniforms would have been retained as the transporter would re-materialize people and clothes as they were from the beam up point.
When Dr. McCoy uses a hypo to gain entrance to Engineering for Scotty and himself: when the door first opens, the shoulder of the set crew can be seen briefly behind the actors.
When the landing party beams back to the correct universe, Spock is waiting for them in the transporter room which is impossible since he could not know the exact time the party was beaming over from the alternate universe.
After Chekov's attempted assassination attempt failed, the crewman that defected from Chekov's crew asks Kirk about his prospects in the captain's crew. Kirk suggests that a commission is in the works, and that he might even make captain. The new henchman smiles with a "yes, sir", then gets punched in the face, with Kirk replying, "Not on my ship." When the camera shows the punched henchman, he's clenching his abdomen in pain, even though he was punched in the face.
The computer records the procedure for getting back to the correct universe on what looks like an old fashioned disk which Kirk removes and hands it to Scotty asking him if he can do it. Without knowing what is on the disk Scotty says he can't do it alone and will need help. Since he has no idea what was recorded, he cannot possibly know if he can accomplish the task nor can he know if he will need help.
Not every crewman salutes Kirk as he and the landing party make their way from the transporter room to Sickbay.
During the fight in sickbay, Kirk, McCoy, Scotty, and Uhura all attack mirror Spock individually. All four are well aware that Humans are physically no match for Vulcans, due to a Vulcan's much greater strength. All four fight for several minutes until Uhura gives Kirk a skull that he uses to bash Mirror Spock on the head. The four could have won the fight much more quickly and easily if they had all surrounded and attacked MirrorSpock simultaneously.
In the final scene, when BarBara Luna's character (Marlena) Capt Kirk a report to sign, he doesn't sign it.
On the bridge, at the end of the story, Kirk and Spock speculate about the whereabouts of the Mirror Kirk and company, with Spock stating that they probably returned to their own universe. Kirk responds that they probably did, but it seems strange that no-one checked the brig to make sure if they did or not.
When the 4 crew are about to return to the transporter Scottie says the controls are set for automatic and Kirk says one of us will have to stay (to operate the controls manually). Scottie says " I will stay Captain" To which Kirk replies : "Get to the transporter Jim". Scotties real name is James Doohan (Jim Doohan). His character is Montgomery Scott. William Shatner therefore made a name error.