Interesting thought experiment. I see many overlap with existing questions from philosophy (of meaning and language), and I think at its core this question is a philosophical one, not about physics.

  The aliens’ language does include words for "left" and 
  "right" and for "left-handed screw" and "right-handed 
  screw". But in all our communications with them, we were 
  never able to figure out which was which. We need them to 
  explain to us which of their words corresponds to our 
  word "left," and which to our word "right." To do so, 
  they will need to appeal to some common point of 
  reference or phenomenon.

Eerily follows the Twin Earth Thought Experiment and its conclusion:

  Yet, at least according to Putnam, when Oscar says water,
  the term refers to H2O, whereas when Twin Oscar says 
  'water' it refers to XYZ. The result of this is that the 
  contents of a person's brain are not sufficient to 
  determine the reference of terms they use, as one must 
  also examine the causal history that led to this 
  individual acquiring the term. (Oscar, for instance,
  learned the word 'water' in a world filled with H2O, 
  whereas Twin Oscar learned 'water' in a world filled with 
  XYZ.)
  ...
  Meanings' just ain't in the head.

This is at the core of semantic externalism. Relating Putnam's view to the OP question:

What if I don't know the difference between left or right? Maybe a brain lesion caused me to skew the concept so that everything would be "left" to me. When I say "move left to avoid hitting a wall" out of this ignorance, this statement can be true or false. If it is true or false, doesn't depend on my flawed concept of "left" or "right", but on those who hear my statement, and correctly know the difference between "left" and "right" and notice the wall on the left side. The true meaning of "Left" is filled in by the language community.

- This could also mean that our community concept of "Left" is objectively false or obsolete. We deem it to be true and meaningful, the aliens don't.

- Or that "Turn Left" could mean "Turn Right" when some time has passed (a dynamic rigid designator?)

- Or that the researchers that communicate with the aliens have a different concept of "Left" than the language community majority.

- Or that their concept of "Left" is simply incommensurable with our concept of "Left". If here on earth scientists "think differently, speak a different language, live in a different world.", imagine how badly alien thoughts, language and experience would translate, if at all.

- If it is true that the structure of language affects the way speakers conceptualize the world, then that would mean that to learn "Alien Left", one would need to learn (some say experience) the entire alien language.

All these things an alien should take into account when entering our language field.

  So they begin to beam information to us in the form of a 
  3-D technical drawing, encoded as a set of 3-D 
  coordinates in space.

A (probably too) simple solution would be for the aliens to send us a 3-D model along with time data. Observing the resulting simulation already implies a perspective and we could reference the first state of the model.