A major chord on the open strings
Open E tunes the six strings to an E major chord: E B E G# B E from the low string up. Struck open it rings as E major, and because a straight bar moves every string together, the bar plays a major chord at every position: A at fret 5, B at fret 7, E again at fret 12.
That simplicity is the point and the limit. A bar or a slant spells five chord types here, the major and minor triads, the power chord, the dominant seventh and the 6sus2, against C6's eight. The sixth and minor seventh sounds C6 is built around are not under the bar.
The retune
Five strings move from C6, and only string 1, the high E, stays. String 6 moves four semitones up, from C to E, and string 5 five semitones down, from E to B. Luna Guitars describes the tuning as open D a whole step higher: D A D F# A D raised two semitones on every string.