diff --git a/Intersections_2/include/CGAL/Intersections_2/Segment_2_Segment_2.h b/Intersections_2/include/CGAL/Intersections_2/Segment_2_Segment_2.h
index d912b2faf30..19cfb0c0548 100644
--- a/Intersections_2/include/CGAL/Intersections_2/Segment_2_Segment_2.h
+++ b/Intersections_2/include/CGAL/Intersections_2/Segment_2_Segment_2.h
@@ -34,6 +34,22 @@ namespace Intersections {
 
 namespace internal {
 
+// indices of lexicographically smallest endpoints
+// depending on config parameter of S2S2_inter_info.
+// {s1_id0,s1_id1,s2_id0,s2_id1}
+constexpr int s2s2_id[8][4] =
+{
+//  seg1   seg2
+  { 0,1,   0,1 },
+  { 0,1,   1,0 },
+  { 1,0,   0,1 },
+  { 1,0,   1,0 },
+  { 0,1,   0,1 },
+  { 1,0,   0,1 },
+  { 0,1,   1,0 },
+  { 1,0,   1,0 }
+};
+
 // struct used to report the combinaric of the intersection
 // of 2 2D segments.
 // More information could be gathered if exposed in do_intersect.
@@ -43,9 +59,20 @@ struct S2S2_inter_info
   bool inter = false;
   bool dim = 0;
   std::array<int, 2> pt_ids = {-1,-1};
+  // integer in [0,7] indicating segment endpoint ordering for determinism
+  // 0: p0 < p1 - p2 < p3 - p0 < p2
+  // 1: p0 < p1 - p3 < p2 - p0 < p3
+  // 2: p1 < p0 - p2 < p3 - p1 < p2
+  // 3: p1 < p0 - p3 < p2 - p1 < p3
+  // 4: p2 < p3 - p0 < p1 - p2 < p0
+  // 5: p2 < p3 - p1 < p0 - p2 < p1
+  // 6: p3 < p2 - p0 < p1 - p3 < p0
+  // 7: p3 < p2 - p1 < p0 - p3 < p1
+  int config;
 
-  S2S2_inter_info(bool inter)
+  S2S2_inter_info(bool inter, int c=-1)
   : inter(inter)
+  , config(c)
   {}
 
   // intersection is an input endpoint
@@ -77,7 +104,7 @@ seg_seg_do_intersect_crossing(
         const typename K::Point_2& p1, const typename K::Point_2& p2,
         const typename K::Point_2& p3, const typename K::Point_2& p4,
         int /* i1 */, int i2, int i3, int /* i4 */,
-        const K& k, bool extra_test)
+        const K& k, bool extra_test, int config)
 {
     switch (make_certain(k.orientation_2_object()(p1,p2,p3))) {
     case LEFT_TURN:
@@ -89,7 +116,7 @@ seg_seg_do_intersect_crossing(
         case RIGHT_TURN:
           return S2S2_inter_info(false);
         case LEFT_TURN:
-          return S2S2_inter_info(true);
+          return S2S2_inter_info(true, config);
         default:
           CGAL_unreachable();
       }
@@ -101,7 +128,7 @@ seg_seg_do_intersect_crossing(
         case COLLINEAR:
           return S2S2_inter_info(i2);
         case RIGHT_TURN:
-          return S2S2_inter_info(true);
+          return S2S2_inter_info(true, config);
         case LEFT_TURN:
           return S2S2_inter_info(false);
         default:
@@ -138,7 +165,7 @@ seg_seg_do_intersect_contained(
         const typename K::Point_2& p1, const typename K::Point_2& p2,
         const typename K::Point_2& p3, const typename K::Point_2& p4,
         int /* i1 */, int /* i2 */, int i3, int i4,
-        const K& k, bool extra_test)
+        const K& k, bool extra_test, int config)
 {
     switch (make_certain(k.orientation_2_object()(p1,p2,p3))) {
     case LEFT_TURN:
@@ -148,7 +175,7 @@ seg_seg_do_intersect_contained(
         case COLLINEAR:
           return S2S2_inter_info(i4);
         case RIGHT_TURN:
-          return S2S2_inter_info(true);
+          return S2S2_inter_info(true, config);
         case LEFT_TURN:
           return S2S2_inter_info(false);
         default:
@@ -164,7 +191,7 @@ seg_seg_do_intersect_contained(
         case RIGHT_TURN:
           return S2S2_inter_info(false);
         case LEFT_TURN:
-          return S2S2_inter_info(true);
+          return S2S2_inter_info(true, config);
         default:
           CGAL_unreachable();
       }
@@ -215,8 +242,8 @@ do_intersect_with_info(const typename K::Segment_2 &seg1,
     typename K::Compare_xy_2 compare_xy;
 
   // first try to filter using the bbox of the segments
-    if (less_xy(A2,B1)
-     || less_xy(B2,A1))
+    if (certainly(less_xy(A2,B1))
+     || certainly(less_xy(B2,A1)))
         return S2S2_inter_info(false);
 
     switch(make_certain(compare_xy(A1,B1))) {
@@ -229,14 +256,14 @@ do_intersect_with_info(const typename K::Segment_2 &seg1,
         case LARGER:
             switch(make_certain(compare_xy(A2,B2))) {
             case SMALLER:
-                return seg_seg_do_intersect_crossing(A1,A2,B1,B2, A1_id,A2_id,B1_id+2,B2_id+2, k, extra_test);
+                return seg_seg_do_intersect_crossing(A1,A2,B1,B2, A1_id,A2_id,B1_id+2,B2_id+2, k, extra_test, (seg1_is_left_to_right ? 0:2) + (seg2_is_left_to_right ? 0:1) );
             case EQUAL:
                 // A1 < B1 < B2 = A1
                 if (extra_test && k.collinear_2_object()(A1, A2, B1))
                   return S2S2_inter_info(B1_id+2, B2_id+2); // DI_MORE_INFO_TAG: A2==B2 but only B2 is reported
                 return S2S2_inter_info(A2_id); // DI_MORE_INFO_TAG: A2==B2 but only A2 is reported
             case LARGER:
-                return seg_seg_do_intersect_contained(A1,A2,B1,B2, A1_id,A2_id,B1_id+2,B2_id+2, k, extra_test);
+                return seg_seg_do_intersect_contained(A1,A2,B1,B2, A1_id,A2_id,B1_id+2,B2_id+2, k, extra_test, (seg1_is_left_to_right ? 0:2) + (seg2_is_left_to_right ? 0:1));
             default:
               CGAL_unreachable();
             }
@@ -275,14 +302,14 @@ do_intersect_with_info(const typename K::Segment_2 &seg1,
         case LARGER:
             switch(make_certain(compare_xy(B2,A2))) {
             case SMALLER:
-                return seg_seg_do_intersect_crossing(B1,B2,A1,A2, B1_id+2,B2_id+2,A1_id,A2_id, k, extra_test);
+                return seg_seg_do_intersect_crossing(B1,B2,A1,A2, B1_id+2,B2_id+2,A1_id,A2_id, k, extra_test, 4 + (seg1_is_left_to_right ? 0:1) + (seg2_is_left_to_right ? 0:2));
             case EQUAL:
                 // B1 < A1 < A2 = B2
                 if (extra_test && k.collinear_2_object()(B1, A1, B2))
                   return S2S2_inter_info(A1_id, A2_id); // DI_MORE_INFO_TAG: A2==B2 but only A2 is reported
                 return S2S2_inter_info(A2_id); // DI_MORE_INFO_TAG: A2==B2 but only A2 is reported
             case LARGER:
-                return seg_seg_do_intersect_contained(B1,B2,A1,A2, B1_id+2,B2_id+2,A1_id,A2_id, k, extra_test);
+                return seg_seg_do_intersect_contained(B1,B2,A1,A2, B1_id+2,B2_id+2,A1_id,A2_id, k, extra_test, 4 + (seg1_is_left_to_right ? 0:1) + (seg2_is_left_to_right ? 0:2));
             default:
               CGAL_unreachable();
             }
@@ -366,15 +393,22 @@ Segment_2_Segment_2_pair<K>::intersection_type() const
     }
 
     // segments intersect in their interiors
-    typename K::FT s1_dx = _seg1->point(0).x() - _seg1->point(1).x(),
-                   s1_dy = _seg1->point(0).y() - _seg1->point(1).y(),
-                   s2_dx = _seg2->point(1).x() - _seg2->point(0).x(),
-                   s2_dy = _seg2->point(1).y() - _seg2->point(0).y(),
-                   lx    = _seg2->point(1).x() - _seg1->point(1).x(),
-                   ly    = _seg2->point(1).y() - _seg1->point(1).y();
+    int c = inter_info.config;
+    std::array<typename K::Point_2, 4> pts = c < 4
+                                           ? CGAL::make_array( _seg1->point(s2s2_id[c][0]), _seg1->point(s2s2_id[c][1]),
+                                                               _seg2->point(s2s2_id[c][2]), _seg2->point(s2s2_id[c][3]) )
+                                           : CGAL::make_array( _seg2->point(s2s2_id[c][2]), _seg2->point(s2s2_id[c][3]),
+                                                               _seg1->point(s2s2_id[c][0]), _seg1->point(s2s2_id[c][1]) );
+
+    typename K::FT s1_dx = pts[0].x() - pts[1].x(),
+                   s1_dy = pts[0].y() - pts[1].y(),
+                   s2_dx = pts[3].x() - pts[2].x(),
+                   s2_dy = pts[3].y() - pts[2].y(),
+                   lx    = pts[3].x() - pts[1].x(),
+                   ly    = pts[3].y() - pts[1].y();
 
     typename K::FT alpha =  (lx*s2_dy-ly*s2_dx)/(s1_dx*s2_dy-s1_dy*s2_dx);
-    _intersection_point = K().construct_barycenter_2_object()(_seg1->point(0), alpha, _seg1->point(1));
+    _intersection_point = K().construct_barycenter_2_object()(pts[0], alpha, pts[1]);
 
     return _result;
 }