I've had to reverse engineer several existing complex data sets. The most important thing to establish are the keys and dependencies in the data. The problem is frankly NP hard, so some intuition and inspection will go a long way to getting you to a sensible answer, so don't count on a simple "turn the crank" solution unless you've got a lot of time on your hands.
What you need to do is to query the data a column at a time and by combinations of columns. You want frequency distributions for column values (and column combination values). Columns (or combos) with maximum frequencies of 1 are candidate keys.
You can also look at frequency distributions of combinations of columns to find potential hierarchies. In your example each value in Col2 only ever has one value in Col1 and so forth.
When you identify candidate keys and dependencies between columns you can apply normalization.
EDIT: In response to OP's quesiton in comments:
This is a query that would tell you whether or not one column may have a functional dependency on another column:
select COL2
from MYSTERY
group by COL2
having count(distinct(COL1)) <> 1
If COL2 has a functional dependency on COL1 then this query would return 0 rows. This is because every value of COL2 has exactly 1 corresponding value of COL1.