How many numbers can we make?
Brachylog, 26 bytes
{{|{|↔}~c₂l₁ʰ↰ᵗc}{+|×}ˡ}ᶜ¹

Try it online!
Times out on test cases of length 5 or greater.
It can probably be golfed further; the first half feels clunky.
Explanation
{…}ᶜ¹  Count unique outputs for predicate.

{|{|↔}~c₂l₁ʰ↰ᵗc}  First part: permute list by reversing and recursing.
{|             }  Either return input unchanged, or
  {|↔}            possibly reverse it,
      ~c₂         split it into two parts,
         l₁ʰ      check that the first part has length 1,
            ↰ᵗ    call this sub-predicate on the second part,
              c   and plug the first part back.

{+|×}ˡ  Second part:
{   }ˡ  left fold by
 +|×    addition or multiplication (chosen separately for each element).