Sounds like it’s this VideoGuide thing, but if you have a minute maybe take another picture with the IR emitters in better focus.
Here’s a link to an old gadget site that describes the VideoGuide unit
I have something almost exactly like that, for ReplayTV. An IR emitter that allows the replay to change channels on the cable bos. You stick one of the IR emitters onto your cable box. Replay calls it an “IR Blaster”. I suspect Tivo has a similar gadget.
I have this gizmolic thing hooked up to my VCR right now. It comes out from the back of the TV and sticks to the infrared sensor on the VCR. The TV can then be programmed to turn on the VCR, start recording, stop recording then turn off the VCR. (RCA TV with guidestar plus.)