Sounds wrong to me, and I would say, "I turned the computer off, and then on again", but I was wondering if both forms are acceptable.
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FumbleFingers' comment points to what's going on here:
In step on and off the scales both on and off act as prepositions, taking scales as their respective objects, and conjoining the two prepositions implies an ellipsis of two conjoined preposition phrases: on the scales and off the scales. This is consonant with how we express the result of each action:
But in turned on (or off) the computer*, on (or off) does not act as a preposition but as an adjective designating a state which you cause the computer to assume. Note that with turn, unlike with step, on (or off) can be posed after the object:
The preposition phrases implicit in the conjunct preposed prepositions are not valid. Consequently we must postpose the conjunction:
I think you're mixing two questions. Turning the computer off and on is something you can only do when it is on to start with. I flicked the light switch on and off (to test the power, because it started in the 'off' position) is a perfectly normal phrase, but does not denote the same process as turning the light off and on (to give the starter a better chance), though that is equally normal.
And whether you insert again in the phrase is largely a matter of choice. It may be that 'flicking the switch on and off' is a single action ending in the off position, where 'flicking it on and off' is more aimless; but equally that may be more precise than most people's language actually is.