Revealing mistake: When Rimmer stands up from underneath the table, just before he says, "Thanks for your support" to Lister, listen very carefully and you can hear the stunt director saying "Action!" backwards. This shot was the splice point between the reversed and the regular footage. (I know this mistake is listed as being in the 'backwards version' on the DVD, but it is in the regular broadcast version too - you just have to listen harder for it.)
Timeslides - S3-E5
Continuity mistake: When we see the skiers, no one is standing in front of the screen. In the next shot, Lister has appeared from nowhere. (00:07:20)
Continuity mistake: From the model shot of Blue Midget, Kryten is seen sitting on the right. When the camera cuts inside the ship, he is sitting on the left. (00:22:40)
Suggested correction: When the exterior shot of Blue Midget is shown, it is not very easy to tell the crew apart. They appear blurred, so it is impossible to determine who is sitting in which place.
Other mistake: When Lister inadvertently sets off the Auto Destruct sequence, the computer voice says the ship will detonate in 15 minutes. The computer then says "the ship will detonate" every five seconds followed by how much time until detonation, but the time in which the computer says this is inconsistent. Two seconds after the computer says 15 minutes, the computer says the ship will detonate in 14 minutes 55 seconds. Ten seconds after saying that, the computer says the ship will detonate in 14 minutes 50 seconds. (00:01:50)
Revealing mistake: After landing on Earth, Rimmer and Kryten are standing on land. When Kryten asks "What is this place?", a car can be seen driving forwards in the far background, even though everything in this version of Earth is supposed to work backwards. (00:06:15)
Other mistake: It is established in the backwards version of Earth, that all writing is reversed and is spelled backwards. However, there are some inconsistencies, as in some instances the writing has been mirrored to appear reversed, whilst in some instances the writing has just been spelled out backwards. For example, car number plates and signs on display in the background have been mirrored to appear reversed, but the newspaper that Kryten reads, the Nodnol sign and the Sensational Reverse Brothers poster appear normally typed out, and not as the mirror image that other aspects of this world have.
Plot hole: The barroom tidy scene in the pub makes absolutely no sense. It is established that this fight started because Lister and the Cat uneat a man's pie, which causes the man to punch Lister, and then the barroom brawl happens, but the events are reversed due to this version of Earth being set in a backwards reality. However, if the events of this fight are put into chronological order, this would mean that Lister uneating the man's pie and being punched by him would have been the last thing to happen during this fight, as it was the first thing to happen in reverse order, and therefore, would also means that there would have been no real reason for the pub patrons would get into a brawl in the first place, as nothing had occurred at that time. (00:23:15)
Plot hole: When Holly discusses what the crew can do as a result of activating the self-destruct, she comes up with three options which all end with "And get blown up." Since it turns out that Holly disconnected the bomb "ages ago," wouldn't she have known that they wouldn't be blown up?
Suggested correction: It's been a while since I've seen the show, but Holly is more than a little eccentric and silly, and if memory serves, it's implied he/she has gone a bit "senile"/glitchy due to sitting around mostly unused for three million years while Lister was in stasis. From the sound of it, it's likely just that Holly was being a bit thick and forgot in the moment about disarming the bomb.
Continuity mistake: When the group are about to open the door with the heat seeking missiles inside and the Polymorph sneaks up behind them, Lister is standing at the front of the group, Cat and Kryten are behind Lister and Rimmer is standing at the back, behind all three. In the next shot, as the missiles exit the door, Rimmer is now suddenly standing front of Cat. (00:24:40)
Revealing mistake: When Rimmer is piloting Starbug in Lister's body, there is a shot of Starbug spinning in circles. This is a shot that was taken from S3 E2 Marooned, of Starbug crashing after colliding with the asteroid. The only difference is the planet in the background has been changed. (00:22:10)
Continuity mistake: When Lister (in Rimmer's body), Cat and Kryten are chasing Rimmer in Blue Midget, the exterior shot shows all three sitting in one row. From the interior, Cat is sitting in front of Lister and Kryten, and Lister and Kryten are sitting behind Cat. (00:22:40)
Continuity mistake: When Rimmer (in Lister's body), is driving Starbug through the moon, the others are following in Blue Midget. One shot shows Blue Midget directly behind Starbug, but another shot shows Blue Midget much further to the right. (00:23:10)
Timeslides - S3-E5
Continuity mistake: When Kryten notices the photo of the bike in the countryside is moving, the peg attached to it is near the middle, along the top, but in the shot from behind, the peg is much closer to the right. The same goes for the photo of Kryten's birthday party. (00:03:30)
Revealing mistake: During the chase on the moon, there is a shot from in front of Starbug crashing into an asteroid. About twenty seconds after this, the exact same shot of Starbug crashing into an asteroid is been repeated a second time, when the ship crashes. (00:23:20)
Timeslides - S3-E5
Continuity mistake: When Lister and Gilbert are looking at the statue of Lister in the courtyard, Lister has his arms crossed in an angle from above, but when the shot cuts back, his arms are at his side, before crossing them again. (00:19:25)
The Last Day - S3-E6
Audio problem: When Lister tries to shoot Hudzen, Hudzen grabs Lister and throws him, but a punch sound effect can be heard, despite no punch being made. (00:26:15)
