How to rotate n individuals at a dinner party so that every guest meets every other guests

I think the answer is:
Make one person stationary, rotate all the folks. This actually works.

Example with N=6. 1 is stationary. (Each person in the top row "meets" the corresponding person just below in the bottom row.)

1) (#1:4) (#2:5) (#3:6) (#4:1) (#5:2) (#6:3)

1 2 3
4 5 6

2) (#1:4,5) (#2:5,3) (#3:6,2) (#4:1,6) (#5:2,1) (#6:3,4)

1 4 2
5 6 3

3) (#1:4,5,6) (#2:5,3,4) (#3:6,2,5) (#4:1,6,2) (#5:2,1,3) (#6:3,4,1)

1 5 4
6 3 2

4) (#1:4,5,6,3) (#2:5,3,4,6) (#3:6,2,5,1) (4:1,6,2,5) (5:2,1,3,4) (6:3,4,1,2)

1 6 5
3 2 4

5) (#1: 4,5,6,3,2) (#2:5,3,4,6,1) (#3:6,2,5,1,4) (4:1,6,2,5,3) (5:2,1,3,4,6) (6:3,4,1,2,5)

1 3 6
2 4 5

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