When shining a red diode laser ($\lambda = 635$ nm) laser through a diffraction grating (5276 lines/cm), I'm observing what I consider to be anomalous maxima near each of the expected maxima for the laser/diffraction grating combination. (See the pictures below.) I used this grating in an introductory physics lab on double slit interference and diffraction gratings. I was unable to explain the anomalous maxima, and simply told my students to ignore them for the purposes of the lab (and they got good results (i.e. the spacing of the maxima, along with the known wavelength of the laser, allowed them to determine the lines/cm of the grating to a reasonable accuracy) by doing this). But, I'm hoping that someone here might be able to explain where these spots are coming from, or at least hypothesize why they're showing up. The grating is fairly old (I'm guessing 20 or so years), and it could probably use some cleaning, but I also observed this effect with a newer grating, which appears to be clean, at least to my eyes. Any ideas what is likely to be causing this?