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More specifically, how is orientation affected with regards to the linear transformations? Does rotation preserve orientation?

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    What does orientation mean to you, usually? – 2017-02-22
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    That's what I'm asking, I don't really have a good reference – 2017-02-22
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    I suppose I meant to ask: since you ask about 2-D specifically, do you know a 3-D definition? – 2017-02-22
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    No, I'm currently interested in regards to planar figures but I will be more than happy to receive a definition for both – 2017-02-22
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    OK. I just wanted to clarify if we were talking about orientation as a property of surfaces, and we are not. Rats. – 2017-02-22

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It sounds like you are asking about projective geometry and it's relationship to geometric transformations. Here's the thing, in projective geometry an object can have some basic transformations done to it. An object can be translated, reflected, rotated, and scaled.

Translation moves the objects coordinates. This means it preserves angles, lengths and orientation. Location isn't preserved.

Rotation is the change of an object via rotation about a specific point. this preserves angles and lengths, but not orientation.

Reflections are the flipping of an object by via a half plane. This preserve angles and lengths, but again not orientation.

Scaling a object is the stretching of an object at a given distance. This rarely preserves anything.

What orientation is, is the position of the angles and points of the object. So if you rotate an object around a given point, you change it's orientation. I hope this helps.