Okay, okay, here's my motivation. I noticed that I can see the LED from remote controls - which should have a peak at 940 nm or so, but also a considerable bandwidth. I tested this as well on a LED I bought that is certainly a 940 nm LED; it's easily visible in the dark and glows a dim red. This is a much longer wavelength than I thought we had any sensitivity to. I'm curious to see if this extends to IR light that's extremely monochromatic, or if I'm just seeing the extreme short-wavelength tail of the LED light that's being emitted. This is why I'd want to use a laser. If 980 is visible in any capacity, I'd find that an interesting result.
Once the laser arrives (probably next week sometime), I will start doing some tests. Actually looking into the beam directly is something I don't really plan on doing, but I wanted to get everybody's take on what would happen if it did shine into my eye. If I don't see anything from diffuse reflections, I am likely to try things like looking at the beam off of a somewhat reflective surface (e.g. coarse metal), and I also may try direct viewing of the second-order maxima from a diffraction grating, or at the worst, reflections from glass surfaces. Ultimately, I'm interested in seeing the minimum power that produces a visual response. Also, the laser beam is focusable and will stay unfocused through the experiment.
weeba2kv - How did you end up with permanent eye damage from a 5 mW laser? Did it take a very long exposure or some other unusual condition? I suppose why I think I'm safe even if I were to shine it into my eye is that there are millions of cheap 5 mW red laser pointers and no cases in the literature of permanent eye damage - even temporary eye damage seems to happen only if people force their eyes open and stare for half a minute or longer (negating the main difference between red and IR exposure - the blink reflex). Keep in mind that even if I try the dumbest (OP) version of this experiment, it would be unfocused and limited to 1-2 seconds at the absolute worst. So if you have had permanent damage from 5 mW, or if anybody knows of any cases like that, it would be good to know about it.
As for the laser being over spec, I suppose I find it unlikely that it would be, say, 20 mW instead of 5. It's supposed to be class IIIa, so that would run afoul of labeling requirements, and wouldn't it be more expensive for them to produce a higher output anyway? If I don't get my hands on a meter, I'll at least compare it to my 940 nm LED and a 780 nm laser which I know to be dim by viewing the spot against a wall with a digital camera, to get some crude qualitative idea as to the power output.
Anyway, I'm not surprised that proposing that I ignore the #1 rule of laser safety would get me flamed. I would just like to state that I'm not completely ignorant of what lasers do, and thanks to everybody who did post helpful replies. Once I get the laser, I'll do some tests and come back here to post what I saw, so anybody interested can know whether or not 980 nm light is completely invisible. And although I find it unlikely, if I end up hurting myself, I'll post that too, so you can all give me the visual equivalent of a Darwin Award.