In quantum field theory (and in string theory), the standard framework for thinking about these things, is in terms of C (charge), P (parity), and T (time) transformations. C swaps particles with their antiparticles, P is a reflection (swaps left and right), and T is time reversal. There is a famous theorem, the CPT theorem, which says that a quantum field theory is "invariant" under the combined CPT transformation (in which all three of these changes are applied simultaneously). So if your theory allows a particular physical process, it must also allow the CPT-transformed counterpart.
Unfortunately this is a part of physics where my understanding is pretty shallow. If I worked through a few of the CPT proofs, and really thought about what was going on, I might have more to say. One of the complications is that defining C, P, and T is not always straightforward. You have to define algebraically how they act on all the observables of the theory under consideration - how all the various types of field (scalar, spinor, vector, tensor...) transform. In string theory there is the additional complication that CPT "inside the string" (on its worldsheet) is a different thing to CPT in the space-time that it moves through.
So I can look up papers where they talk about CPT, but for me it's just opaque algebra - do all these transformations, and a certain thing happens. In particular, I haven't made that intuitive connection between the algebra, and my private visualizations of reflection and time reversal, etc - the price of not having properly worked through it. In turn that means I can't fluently talk about the similarities and differences between a naive concept of reflection or time reversal, and the technical concept that is given that name. Hopefully in time I'll get there. But meanwhile I decided to jump in anyway and respond to this post, because it was just sitting there unanswered...
Regarding the orientation of a string, it is not defined by anything like a physical flow from one point to another. You can certainly have e.g. a wave traveling along or around a string in a particular direction, but the string's "orientedness" is not due to the presence of anything like that, it doesn't require it. It's more that, in a so-called orientable space, you are able to define a difference between left and right, or up and down. You can switch the labels around, but there's still an absolute difference between one direction and the other.
In terms of C, P and T as they are technically defined, I would think that time reversal, T, would not reverse the spatial orientation of a string. Instead, reversing the orientation of a string would correspond to P, reflection, the "parity transformation" that swaps left and right. Meanwhile, it's the combined transformation, CPT, which will actually give you the physical mirror process to what you started with (the "anti-process" which is also a solution of your theory); so along with T and P, you need to apply C, and invert any charges that the string may be carrying (whether they are located on its ends, as in an open string theory, or smeared around it, as in the heterotic string).
