I recently found that the chord progression in Tom Paxton's song I Don't Want Your Pardon is rather weird. The song is in the key of E.

Apart from itself, common chords in E major songs would include A, B7, G#m, C#m, F#m, and sometimes maybe Am, F# and G#. But this Paxton song uses none of these chords. Instead, it has the following chord progression:

E - C - E
E - C - D - E
E - G - E
E - G - D - E

The chord progression is very simple, not complicated at all - but are all unusual chords for a song in E major. How come they sound smooth together?