Common Pentatonic Pairings
The ordinary, common-as-muck pentatonic scale gets much more interesting when you move it around.
Playing minor pentatonic in root position gives you the notes 1 b3 4 5 b7. You can shift it to any other pitch you like, but the results are not all equal. For example, shifting up by a whole tone gives you, in combination with the original notes, the Dorian scale; shifting down by the same amount gives the Phrygian. Not very exciting.
Shifting up by a minor third is a useful trick, but it's really just the minor up a minor third trick with a few notes missed out. Shifting down by a minor third gives us major pentatonic, which of course is also a good sound but a very familiar one.
Shifting up or down by a fourth produces pentatonics that overlap with the original by all but one note; these are not worth paying special attention to, at least in my opinion.
This leaves five possibilities.
Shifting up or down by a semitone is a well-known technique among jazz musicians known as "side-slipping". The two actually sound very different. And shifting by a tritone gives a good sound, too. The crucial thing about these three is that the shifted scale has no overlaps with the original, so mixing up their notes provides ten of the twelve in the total chromatic.
Here are fingerboard diagrams that might help you "see" these -- the blue scale is the original, the yellow/green one the shifted one (dark blue and green are their respective roots):
Shifting up or down by a major third produces a stranger effect. The former gives us 2 3 5 6 7, which doesn't look too bad but when combined with the original produces 1 2 b3 3 4 5 6 b7 7, a nonatonic scale with a lot of semitones. It's extremely dissonant, and of course shifting down by a major third instead just produces a mode of the same scale. The coscale is a sus 4 triad, with its root a semitone above the original pentatonic -- for those keeping score at home, that means you play A minor pentatonic plus C# minor pentatonic and get Bb sus 4 as the coscale.
Note that the two common pentatonics separated by a major third share one note, which is the 5 of the lower one and the b3 of the upper one. This is useful for fretboard visualisation and also for pivoting melodically from one to the other.
I use the ones in the diagram above a lot. Pentatonics sound so familiar that you can generate lines with a lot of dissonance in them and still "sell" them as consonant using this technique. I don't know what to make of the major-third-shifted ones though; maybe I'll figure out what to do with them eventually.
Incidentally, there are two more Common Pentatonic hypermodes: one comes from shifting minor pentatonic up a minor third, and the other comes from shifting it down a major second. The former is just a flavour of my minor up a minor third hypermode trick. The other is, like the major third shifts, less clear to me at this point. One way to look at it is as a subset of Phrygian or Locrian in the original key -- that is, on A minor you can play G minor pentatonic, implying A Phrygian or A Locrian.