diff --git a/src/sage/tensor/modules/comp.py b/src/sage/tensor/modules/comp.py
index ad60a401a99..7c14863b1af 100644
--- a/src/sage/tensor/modules/comp.py
+++ b/src/sage/tensor/modules/comp.py
@@ -2123,6 +2123,34 @@ def contract(self, *args):
             sage: a.contract(0, a, 0) == b.trace(0,1)
             True
 
+        TESTS:
+
+        Check that :trac:`32355` is fixed::
+
+            sage: from sage.tensor.modules.comp import CompFullyAntiSym
+            sage: a = CompFullyAntiSym(QQ, V.basis(), 2)
+            sage: a[0,1] = 1
+            sage: b = CompFullyAntiSym(QQ, V.basis(), 2)
+            sage: b[0,1], b[0,2] = 2, 3
+            sage: a.contract(0, 1, b, 0, 1)
+            4
+            sage: a.contract(0, 1, b, 1, 0)
+            -4
+            sage: a.contract(1, 0, b, 0, 1)
+            -4
+            sage: a.contract(1, 0, b, 1, 0)
+            4
+            sage: Parallelism().set('tensor', nproc=2)  # same tests with parallelization
+            sage: a.contract(0, 1, b, 0, 1)
+            4
+            sage: a.contract(0, 1, b, 1, 0)
+            -4
+            sage: a.contract(1, 0, b, 0, 1)
+            -4
+            sage: a.contract(1, 0, b, 1, 0)
+            4
+            sage: Parallelism().set('tensor', nproc=1)  # switch off parallelization
+
         """
         #
         # Treatment of the input
@@ -2164,31 +2192,34 @@ def contract(self, *args):
             # ncontr indices and call the method index_generator() on it:
             comp_for_contr = Components(self._ring, self._frame, ncontr,
                                         start_index=self._sindex)
-            res = 0
 
+            # Pairs of indices tuples for the contraction:
+            ind_pairs = []
+            for ind_s in comp_for_contr.index_generator():
+                ind_o = [None for i in range(ncontr)]
+                for pos_s, pos_o in contractions:
+                    ind_o[pos_o] = ind_s[pos_s]
+                ind_pairs.append((ind_s, ind_o))
 
             if Parallelism().get('tensor') != 1:
-                # parallel contraction to scalar
+                # parallel computation
 
-                # parallel multiplication
-                @parallel(p_iter='multiprocessing',ncpus=Parallelism().get('tensor'))
-                def compprod(a,b):
+                @parallel(p_iter='multiprocessing', ncpus=Parallelism().get('tensor'))
+                def compprod(a, b):
                     return a*b
 
                 # parallel list of inputs
-                partial = list(compprod([(other[[ind]],self[[ind]]) for ind in
-                                     comp_for_contr.index_generator()
-                    ]))
-                res = sum(map(itemgetter(1),partial))
+                partial = list(compprod([(self[[ind_s]], other[[ind_o]])
+                                         for ind_s, ind_o in ind_pairs]))
+                res = sum(map(itemgetter(1), partial))
             else:
-                # sequential
+                # sequential computation
                 res = 0
-                for ind in comp_for_contr.index_generator():
-                    res += self[[ind]] * other[[ind]]
+                for ind_s, ind_o in ind_pairs:
+                    res += self[[ind_s]] * other[[ind_o]]
 
             return res
 
-
         #
         # Positions of self and other indices in the result
         #  (None = the position is involved in a contraction and therefore
@@ -2300,16 +2331,18 @@ def compprod(a,b):
             nproc = Parallelism().get('tensor')
             lol = lambda lst, sz: [lst[i:i+sz] for i in range(0, len(lst), sz)]
             ind_list = [ind for ind in res.non_redundant_index_generator()]
-            ind_step = max(1,int(len(ind_list)/nproc/2))
-            local_list = lol(ind_list,ind_step)
+            ind_step = max(1, int(len(ind_list)/nproc/2))
+            local_list = lol(ind_list, ind_step)
 
             listParalInput = []
             for ind_part in local_list:
-                listParalInput.append((self,other,ind_part,rev_s,rev_o,shift_o,contractions,comp_for_contr))
+                listParalInput.append((self, other, ind_part, rev_s, rev_o,
+                                       shift_o, contractions, comp_for_contr))
 
             # definition of the parallel function
-            @parallel(p_iter='multiprocessing',ncpus=nproc)
-            def make_Contraction(this,other,local_list,rev_s,rev_o,shift_o,contractions,comp_for_contr):
+            @parallel(p_iter='multiprocessing', ncpus=nproc)
+            def make_Contraction(this, other, local_list, rev_s, rev_o,
+                                 shift_o,contractions, comp_for_contr):
                 local_res = []
                 for ind in local_list:
                     ind_s = [None for i in range(this._nid)]  # initialization
@@ -2334,7 +2367,7 @@ def make_Contraction(this,other,local_list,rev_s,rev_o,shift_o,contractions,comp
                 for jj in val:
                       res[[jj[0]]] = jj[1]
         else:
-            # sequential
+            # sequential computation
             for ind in res.non_redundant_index_generator():
                 ind_s = [None for i in range(self._nid)]  # initialization
                 ind_o = [None for i in range(other._nid)] # initialization