With the conscious decision to eschew elegance for reliability, I present

 symSum[li_List] := Module[{k2 = Ceiling[Length[li]/2]}, 
 Total[MapAt[Reverse, 
 If[Apply[Equal, Length /@ #], #, 
 MapAt[Function[l, PadLeft[l, k2]], #, {2}]] &[
 Partition[li, k2, k2, {1, 1}, {}]], {2}]]]

or more compactly,

 symSum[li_List] := Module[{k2 = Ceiling[Length[li]/2]}, 
 Total[MapAt[Reverse, 
 PadLeft[Partition[li, k2, k2, {1, 1}, {}], {2, k2}], {2}]]]

Testing:

 list = {a, b, c, d, e, f};
 
 symSum[list]
 {a + f, b + e, c + d}
 
 symSum[Most@list]
 {a + e, b + d, c}

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Yet another variation:

 symSum[li_List] := Module[{k = Length[li], k2},
 k2 = Ceiling[k/2]; 
 Total[MapAt[
 Composition[Reverse, If[EvenQ[k], Identity, RotateRight]], 
 Internal`Deflatten[PadRight[li, 2 k2], {2, k2}], {2}]]]

and we can keep on putting out variations until we're all blue in the face:

 symSum[li_List] := Module[{k = Length[li], k2},
 k2 = Ceiling[k/2];
 PadRight[Total[
 Take[li, {#, # Quotient[k, 2], #}] & /@ {1, -1}], k2, test[[k2]]]]