randomly join two tables (not Cartesian)

Hi, I have one table but would like to double the information in it
randomly. I know that I can randomize the tables by an order by
rand() clause. However, how can I arbitrarily join them without a
condition, without a Cartesian product? The only condition would be
that row 1 joins with row 1, row 2 with row 2.... but only after
they've been randomized. Thus, all data is used exactly one time per
table and twice in the derived table. Thanks!

For instance if my table is this:
id name
1 Sarah
2 Louis
3 Bobby

Then I want my statement to give me this:
id name id name
1 Sarah 2 Louis
2 Louis 3 Bobby
3 Bobby 1 Sarah