The use of a colon is to name something or give clarifying examples of it in the form of list items.
As for usage rules, Mignon Fogarty discusses them in her article "How to use one punctuation mark: the colon."
The most important thing to remember about colons is that you only use them after statements that are complete sentences. Never use a colon after a sentence fragment.
For example, it's correct to say, "Grammar Girl has two favorite hobbies: watching clouds and seeing how long she can stand on one foot." That's correct because "Grammar Girl has two favorite hobbies" is a complete sentence all by itself.
Notice how the items after the colon expand on or clarify what came before the colon. I referred to my favorite hobbies before the colon and then specifically named them after the colon. A quick and dirty way to decide whether a colon is acceptable is to test whether you can replace it with the word namely. For example, you could say, "Grammar Girl has two favorite hobbies, namely, watching clouds and seeing how long she can stand on one foot." Most of the time, if you can replace a colon with the word namely, then the colon is the right choice.
. . . The rules are the same whether you are writing lists or sentences: use a colon when you could use the word namely and after something that could be a complete sentence on its own.
I suspect that some people relax the rule about sentence fragments when introducing a quotation (I am guilty of that and should pay more attention). If you could replace the colon with a period, and if the text that precedes it could be taken as a complete sentence, then you are fine.
However, here are some examples:
I stopped at a fast food restaurant: McDonald's.
I had four things for breakfast: eggs, bacon, toast, and coffee.
On the other hand, and this is something not mentioned in the article but that I've observed in my own use, the text that follows a colon typically is a sentence fragment; it is simply one or more items. (Although, this is not always the case—as the last sentence of Mignon Fogarty's that I quoted demonstrates.)
Given that, let's turn to your example sentence:
Some shapes are symmetrical: for example, when cutting a circle or a square through the middle, two identical shapes are produced.
As per some of the comments, this reads more like two separate sentences. It makes more sense to me to use a period or a semicolon.
Part of the problem lies in my parsing of it.
Some shapes are symmetrical: circles and squares.
Here, these are two examples of symmetrical shapes. But the remainder of the text after the colon is strangely separated from those list items.
Some shapes are symmetrical: circles and squares. By cutting them through the middle, two identical shapes are produced.
That sounds fine, but now we have two sentences.
If I wanted to maintain everything in a single sentence, I would have to move the last part of the sentence to the start:
Two identical shapes can be produced by cutting symmetrical shapes through the middle: circles and squares.
But I'm not satisfied with any of these versions. In particular, they lose the sense of the now missing "for example" or an alternatively missing "including." In other words, it makes it sound as if circles and squares are the only symmetrical shapes—but that's just not true.
I can take my earlier breakfast example and mention only two food items without implying an exhaustive list:
My breakfast included two food items in particular: eggs and bacon.
With this phrasing, we don't rule out the possibility that there were more than two food items; we simply call attention to two of them. (The in particular could be left out, but it doesn't seem to me that included on its own imparts non-exclusivity explicitly enough.)
Something similar could be done to the example sentence:
Symmetrical shapes that can be cut through the middle and produce two identical shapes include two in particular: the circle and the square.
This seems to address all of my parsing issues; however, it's a lot of effort to make the use of a colon (rather than a semicolon) work—and by changing things around so much, the meaning of the sentence is also altered more than you might like.
Of course, you could just leave your sentence as it is—with the exception of removing for example. It would be perfectly grammatical. But I acknowledge that you don't like that option.
I'm not convinced that there isn't some construction where list items following a colon could include for example. But I don't believe this is one of them.
Lastly, you can just forget all about this and not make any changes at all. I'm sure very few people would be overly concerned by it.
