I don’t think the gestures are specific, except to the individual person. In rehearsal, you get to learn what the conductor’s personal patterns and behaviours are, and figure out the signals.
At the final performance, the conductor is less specifically necessary to be observed (though they are still rigorously followed, and out the corner of your eye you can see when your cues are being signalled) because most of the real work they do happens in rehearsal.
I once had a conductor (this was when I was in high school) teach us all how to conduct. Well, obviously it was only the basics, because it took about twenty minutes. He might have been the best conductor I’ve ever played with, just because of this. It made following his conducting extremely easy.
So, say you’re playing a piece in 3/4 (waltz) time. As this conductor taught us, at the first beat in a measure, your right hand should be up. On the second, you lower it a couple inches, and on the third, a couple inches to the right. Going to the next measure, your hand goes back to the first position. So you’re drawing a little triangle in the air. This helps the musicians know where they are in the measure and how fast to play.
But this is only the barest of bare bones of what a conductor really does. To see the work involved, you’d really need to watch rehearsals. The conductor’s job is to make sure everyone plays together so that there is one unified sound. Two people can play the exact same piece of music correctly - and have it come out completely differently. Although music has commands on it, they can be interpreted in a many different ways. The conductor’s job is to make sure everyone is interpreting them in the same way. My allegro (fast) might be slower than yours. Or your staccatos might not be as staccato as mine. Maybe my idea of piano is so piano that you can’t hear me. The conductor decides what’s the what and makes everyone get along, in a musical sense.
What you see in a performance is a somewhat stylized idea of what a conductor should be doing. In theory, anyway, an ensemble should know the piece well enough before a performance that they don’t really need a conductor. But it’s nice to have, in case you mess up your counting and need a cue.
The conductor is sort of equivalent to the sound guy in a club or a music producer in a studio. He/she is the only one that is in the right position to hear everyone during the performance. Just like a rock guitarist can’t hear what’s coming out of the amps and going to the audience because he’s behind the amps, the trombonist can’t hear what’s coming out of the clarinet section because she’s behind them (obviously they can hear it but not in the same way the audience can).
So, just like the sound guy or producer sits with his big mixing board and moves levels up and down to control the sounds coming out of the instruments and microphones during the performance or recording, the conductor uses his/her arms and hands to control all of the elements of the music coming out of the band members during the performance.
And, not only is the conductor the only one in the right place to hear all the musicians, he/she also is the only one with ALL of the notes in front of them! It’s sort of like Microsoft - no one knows exactly what the other workers are doing, just that they have to make their bit work in concert (ha!) with everyone else’s bits.
Yeah, it shouldn’t be a big mystery.
Basic Conducting Motions (In all it’s Mormon goodness) 
Here’s Sibelius’ Valse Triste*. (This is really a fantastic video)
(0:18) - Note how “loopy” the downbeats are and how it goes with how lazy the music is.
(0:36) - Watch his left hand come up as the dynamic level of the orchestra rises.
(0:36) - His hand goes palm down as the dynamic level falls again.
(0:55) - Watch the head nod. Takes some rubato entering the next phrase.
(1:32) - Things get a bit more “bouncy” as the music changes.
(1:39) - Again, palm down as he wants the dynamic level (and tempo slightly) to decrease.
(2:14) - Rubato section that would be pretty much impossible without a conductor to keep the orchestra together.
(2:45) - Check out the look on his face as the music gets more intense.
(3:40) - Tempo picks up. Would be very hard for everyone to move forward at the same rate without a conductor.
(4:12) - That’s flying. That’s easily the fastest I’ve ever heard that section.
(4:30) - Almost impossible to get such a precise cutoff without someone giving a visual cue.
(4:50) - Really conducts the last four notes with his head. :dubious:
But the point is even if you have never worked with the guy, just his body language should be enough for you to get a ROUGH idea of what he’s looking for.
*Valse Triste has a special place in my heart. When I was in college, once a semester the conducting class got experience conducting a live group…ours. So 20 students * two semesters * 4 years in college…you do the math. 
Maybe I’m not interpreting what you are saying exactly, but I’ve never seen anything like this except for the triangle concept.
Standard pattern for 3/N time would be a downstroke for “one”, inward (to the left for right hand, right for left hand) for “two” and outward (reverse the previous) for “three”, then your hands should be high, ready for the next downstroke.
Note that either hand or both can conduct the beat. The other can duplicate the beat hand (in mirror, usually), or be used for other things, like cueing players or indicating dynamics. Conductors of small groups don’t need both hands as much as large groups as most players/singers can see OK. For large groups, the conductor needs to have large motions that can be seen from all angles.
There’s more to just pointing a hand in a different direction for each beat. People have told me that my conducting is easy to follow, perhaps for two reasons: I like to give a definite “bounce” to each beat (on downstroke, bounce at the bottom) and I tend to anticipate (lead) with my shoulders, body and even eyebrows just before a beat, especially the first one of a piece. If all of those go up, down is likely to quickly follow.
Conversely, I have performed under conductors who are hard to follow. One professional I recall always subdivided each beat with his hands and it was hard to tell exactly what beat or sub-beat or sub-sub-beat he was working on. His hands sort of fluttered around the major divisions.