Factual error: The writer's understanding of the history of Nazism and its leaders is a bit shonky. Claus von Stauffenberg did not plant a bomb in Hitler's briefcase - he put it in his own briefcase which he planted in a meeting room next to Hitler (some berk moved it). This was in July 1944, while the last Nuremberg rally - which Lister visits, bringing back the briefcase - was in 1938. Stauffenberg didn't even join the anti-Hitler conspiracy until 1943. Red Dwarf is not an 'alternate history' - they correctly identify elements of the Stauffenberg plot and the Nazi regime, they just get them wrong.
Plot hole: In the episode 'Backwards' most of the things that happen on 'Backwards Earth' are correct, except, of course, in reverse. But there is an exception: in the scene in the Cafe, when the waitress comes to 'dirty' the table, she tips a box of trash on the table. If you play this scene backwards, the trash leaps from the table into the box. Even on a 'backwards earth' I can't believe that trash spontaneously leaps about...
Continuity mistake: When Lister (in Rimmer's body) pleads the Cat to lend him his body, he says 'how else can I pilot White Midget?'. In the next scene, the Blue Midget is being piloted, and White Midget was originally abandoned for Starbug. (this mistake was admitted by the crew)
Other mistake: At the start of the episode, there is a spoof Star Wars-type caption, which spools up the screen very quickly. Too quickly to be read, in fact... which may explain why few people have noticed that the last line of the caption, "And now the Saga Continuums...", is written twice.
Plot hole: If everything goes backwards, shouldn't Kryten and Rimmer have got fired, done their job and then got hired? Rimmer & Kryten discovered that everything was backwards in the cafe earlier in the episode. At the club they did their act and then got fired. Put this 'forwards' and it becomes wrong.
Revealing mistake: Near the end of the episode, Kryten and his employer are seen arguing in supposedly backwards English. However, when you watch the episode how the people of earth would see it on the second DVD, Kryten and his employer aren't speaking English, they are just speaking plain gobbeldy-gook, even although when you see the people of Earth talking on the DVD they speak correctly.
Continuity mistake: Kryten removes the polymorph underpants and throws them under the covers of Lister's bed. When he checks the bed and pulls out a snake the pants are now on top of the covers (even though he says they're gone) and the covers themselves are now arranged differently.
Continuity mistake: Kryten needs to ask what a pub is, yet in the episode "Backwards" he and Rimmer worked in one for three weeks as the comedy act "The Sensational Reverse Brothers", and, later, were involved in a brawl in the same pub. Originally, the episodes were to be shown in a different order, which explains why the mistake occurs, but doesn't excuse it.
Plot hole: In season 2 episode 2 Better Than Life, Rimmer says his brother Frank was 6 foot 5 by the time he was 11. However, in season 3 episode 5 Timeslides, the picture of Frank Rimmer's wedding shows he is around Rimmer's height which would be about just under 6 foot.
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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.)