Print out a word in a very large font on a piece of paper. Now, hold the blank side up to a light and try to read the word - it will be backwards. This is because you're reading the paper back to front.
Our mind makes it a left to right inversion because we first try to think of it as a rotation. Look in the mirror while holding a hat in your left hand and a shoe in your right (the objects are arbitrary - substitute where necessary). Now, look in the mirror. The hat will be on YOUR left and the shoe will be on YOUR right. There's no left to right swap. The reason that I capitalized "YOUR" is that I don't want you considering whether it's on the left or right of your REFLECTION. That's where the left to right inversion mistake comes from - we think the shoe is on the reflection's left. Normally, when we're facing another person, that person has rotated 180 degrees to face us. That puts her left on our right and vice versus. We mentally assume that's the case with the reflection in the mirror - that the reflection's right hand is on our left. However, the reflection wasn't rotated 180 degrees to face us. It was flipped front to back. The reflection's left hand is on our left, just as you would expect from something that has not been flipped left to right.
Our mind makes it a left to right inversion because we first try to think of it as a rotation. Look in the mirror while holding a hat in your left hand and a shoe in your right (the objects are arbitrary - substitute where necessary). Now, look in the mirror. The hat will be on YOUR left and the shoe will be on YOUR right. There's no left to right swap. The reason that I capitalized "YOUR" is that I don't want you considering whether it's on the left or right of your REFLECTION. That's where the left to right inversion mistake comes from - we think the shoe is on the reflection's left. Normally, when we're facing another person, that person has rotated 180 degrees to face us. That puts her left on our right and vice versus. We mentally assume that's the case with the reflection in the mirror - that the reflection's right hand is on our left. However, the reflection wasn't rotated 180 degrees to face us. It was flipped front to back. The reflection's left hand is on our left, just as you would expect from something that has not been flipped left to right.