Permuations & Inverses Have Coprime Adjacent Pairs

[Leroy Quet]

Sequence A076220 is the number of permutations of (1,2,3,...n) such that 
every adjacent pair of elements is relatively prime.

But what would the sequence be such that, not only is every adjacent 
element pair coprime, but every adjacent pair in each permutation's 
inverse permutation is coprime too?

I get that the sequence begins:
1, 2, 6, 9,...

For example, the 9 permuations of 1,2,3,4 with coprime adjacent pairs and 
whose inverse permutations also have coprime adjacent terms are:

(Only 2 3 4 1 and 4 1 2 3 are not their own inverse.)

1 2 3 4
1 4 3 2
2 1 3 4
2 1 4 3
2 3 4 1   4 1 2 3
3 2 1 4
3 4 1 2
4 3 2 1

Could someone please calculate/submit this sequence if it is not already 
in the EIS?

We could also ask about the number of such permuations if the first and 
last elements must be coprime also.
Here we have the sequence:
1, 2, 6, 8,...

thanks,
Leroy Quet