I agree with you. (It's an odd sentence, isn't it?)
Yes, very odd. You normally don't find 'other' in the opening phrase of a sentence - before the listener knows what it refers to. In this case, I'd expect something like "When one of your eyes is being checked, please close the other".
I want to add a question.
I've learnt that "another" is used only with singular countable nooun because it's simply an+other.
My question is: Is it possible to use "other" with singular countable nouns.
For example: I have otherpen. ????