diff --git a/spot/twaalgos/toparity.cc b/spot/twaalgos/toparity.cc
index b4ab6987b..f612e4ca1 100644
--- a/spot/twaalgos/toparity.cc
+++ b/spot/twaalgos/toparity.cc
@@ -19,10 +19,11 @@
 
 #include "config.h"
 #include <spot/twaalgos/toparity.hh>
+#include <spot/twaalgos/sccinfo.hh>
 
 #include <deque>
 #include <map>
-
+#include <numeric>
 #include <sstream>
 
 namespace spot
@@ -88,15 +89,14 @@ namespace spot
             return it.first->second;
           };
 
+        std::vector<unsigned> initial_perm(aut_->num_sets());
+        std::iota(initial_perm.begin(), initial_perm.end(), 0);
         {
-          std::vector<unsigned> p0;
-          for (unsigned k : aut_->acc().all_sets().sets())
-            p0.push_back(k);
-          lar_state s0{aut_->get_init_state_number(), p0};
-          unsigned init = get_state(s0); // put s0 in todo
-          res_->set_init_state(init);
+          lar_state s0{aut_->get_init_state_number(), initial_perm};
+          res_->set_init_state(get_state(s0));
         }
 
+        scc_info si(aut_, scc_info_options::NONE);
         // main loop
         while (!todo.empty())
           {
@@ -106,6 +106,7 @@ namespace spot
             // TODO todo could store this number to avoid one lookup
             unsigned src_num = get_state(current);
 
+            unsigned source_scc = si.scc_of(current.state);
             for (const auto& e : aut_->out(current.state))
               {
                 // find the new permutation
@@ -118,6 +119,12 @@ namespace spot
                     std::rotate(it, it+1, new_perm.end());
                   }
 
+                if (source_scc != si.scc_of(e.dst))
+                  {
+                    new_perm = initial_perm;
+                    h = 0;
+                  }
+
                 lar_state dst{e.dst, new_perm};
                 unsigned dst_num = get_state(dst);
 
diff --git a/tests/python/toparity.py b/tests/python/toparity.py
index e714a72d7..b0bffbe23 100644
--- a/tests/python/toparity.py
+++ b/tests/python/toparity.py
@@ -35,6 +35,18 @@ State: 0
 p = spot.to_parity(a)
 assert spot.are_equivalent(a, p)
 
+a = spot.automaton("""
+HOA: v1 States: 6 Start: 0 AP: 2 "p0" "p1" Acceptance: 6 Inf(5) |
+Fin(2) | Inf(1) | (Inf(0) & Fin(3)) | Inf(4) properties: trans-labels
+explicit-labels trans-acc --BODY-- State: 0 [0&1] 2 {4 5} [0&1] 4 {0 4}
+[!0&!1] 3 {3 5} State: 1 [0&!1] 3 {1 5} [!0&!1] 5 {0 1} State: 2 [!0&!1]
+0 {0 3} [0&!1] 1 {0} State: 3 [!0&1] 4 {1 2 3} [0&1] 3 {3 4 5} State:
+4 [!0&!1] 1 {2 4} State: 5 [!0&1] 4 --END--
+""")
+p = spot.to_parity(a)
+assert p.num_states() == 22
+assert spot.are_equivalent(a, p)
+
 for f in spot.randltl(4, 400):
     d = spot.translate(f, "det", "G")
     p = spot.to_parity(d)