f([10], [10]) == "e"
f([10, 30, 20], [30, 20, 10]) == "e"
f([10, 30, 20, 40], [30, 20, 40, 10]) == "o"

żṗ2</€⁺Sị“oe

żṗ2</€⁺Sị“oe  Main link. Arguments: A, B (lists)

ż             Zip A with B. Yields an array of pairs [x, σ(x)].
 ṗ2           Generate all pairs [[x, σ(x)], [y, σ(y)]].
   </€        Reduce each pair by </€.
              This maps [[x, σ(x)], [y, σ(y)]] to [x < y, σ(x) < σ(y)].
      ⁺       Repeat the previous link, i.e., execute </€ once more.
              This maps [x < y, σ(x) < σ(y)] to ((x < y) < (σ(x) < σ(y))), which is
              true if and only if x > y and σ(x) < σ(y).
       S      Sum. This counts the number of inversions.
        ị“oe  Retrieve the letter at the corresponding index.
              Indexing is 1-based and modular, so an odd sum retrieves the first
              letter, an even sum the second.

2$St!<Rz2'oe'w)

2$S     % implicitly take two row vectors. Sort second and apply the indices
        % of that sorting to the first
t!      % duplicate. Transpose into column vector
<       % true for elements of the column vector that exceed those of the
        % row vector. Gives a 2D array with all pairs of comparisons
R       % keep only upper triangular part of that array
z       % number of nonzero elements. This is the number of inversions
2      % parity of that number: gives 0 or 1
'oe'w   % push string 'eo' below the top of the stack
)       % apply index to produce 'e' or 'o'. An index 1 refers to the first
        % element, whereas 0 refers to the last. Implicitly display

p=@(v)eye(nnz(v))(v,:);@(a,b)'ole'(det(p(a)*p(b))+2)

(a,b)=>"eo"[r=0,g=a=>a.map((e,i)=>a.slice(i).map(d=>r^=d<e)),g(a),g(b),r]

If[Mod[Plus@@Length/@(Join[{0},#]&)/@PermutationCycles[#][[1]],2]==0,"e","o"]&

If[Tr[Signature/@{##}]==0,o,e]&