Everywhere in the code that was doing things wrong has been changed.

Two interesting points of note:
* The touchspecial sector flag seems to actually do its job now.
* Detection of sectors with polyobjects in seems to have done this incorrectly, but this doesn't mess with anything about touching the polies themselves so it seems to really only handle edge cases where the polyobject was too close to the border of another sector (which would've likely made rendering glitches anyways).
* There was a whole swathe of teetering code that was basically never run properly because of this mistake. I did a simple fix at first, but you started teetering whenever you were slightly less than your radius away from a sector's edge, which was completely different and undesirable behaviour. Instead, I cut out the code that was never running, and just left the hacky method in instead since it was more accurate to what we want in general.

src/p_mobj.c

@@ -2703,7 +2703,7 @@ static void P_PlayerZMovement(mobj_t *mo)
 					msecnode_t *node;
 					boolean stopmovecut = false;
 
-					for (node = mo->touching_sectorlist; node; node = node->m_thinglist_next)
+					for (node = mo->touching_sectorlist; node; node = node->m_sectorlist_next)
 					{
 						sector_t *sec = node->m_sector;
 						subsector_t *newsubsec;
@@ -3648,7 +3648,7 @@ static void P_PlayerMobjThinker(mobj_t *mobj)
 	if (!(netgame && mobj->player->spectator))
 	{
 		// Crumbling platforms
-		for (node = mobj->touching_sectorlist; node; node = node->m_thinglist_next)
+		for (node = mobj->touching_sectorlist; node; node = node->m_sectorlist_next)
 		{
 			fixed_t topheight, bottomheight;
 			ffloor_t *rover;
@@ -3673,7 +3673,7 @@ static void P_PlayerMobjThinker(mobj_t *mobj)
 	{
 		boolean thereiswater = false;
 
-		for (node = mobj->touching_sectorlist; node; node = node->m_thinglist_next)
+		for (node = mobj->touching_sectorlist; node; node = node->m_sectorlist_next)
 		{
 			if (node->m_sector->ffloors)
 			{
@@ -3694,7 +3694,7 @@ static void P_PlayerMobjThinker(mobj_t *mobj)
 		}
 		if (thereiswater)
 		{
-			for (node = mobj->touching_sectorlist; node; node = node->m_thinglist_next)
+			for (node = mobj->touching_sectorlist; node; node = node->m_sectorlist_next)
 			{
 				if (node->m_sector->ffloors)
 				{

src/p_spec.c

@@ -3308,7 +3308,7 @@ sector_t *P_PlayerTouchingSectorSpecial(player_t *player, INT32 section, INT32 n
 		return rover->master->frontsector;
 	}
 
-	for (node = player->mo->touching_sectorlist; node; node = node->m_thinglist_next)
+	for (node = player->mo->touching_sectorlist; node; node = node->m_sectorlist_next)
 	{
 		if (GETSECSPECIAL(node->m_sector->special, section) == number)
 		{
@@ -4657,7 +4657,7 @@ void P_PlayerInSpecialSector(player_t *player)
 		P_RunSpecialSectorCheck(player, sector);
 
 	// Iterate through touching_sectorlist
-	for (node = player->mo->touching_sectorlist; node; node = node->m_thinglist_next)
+	for (node = player->mo->touching_sectorlist; node; node = node->m_sectorlist_next)
 	{
 		sector = node->m_sector;
 

src/p_user.c

@@ -2849,86 +2849,10 @@ static boolean PIT_CheckSolidsTeeter(mobj_t *thing)
 //
 static void P_DoTeeter(player_t *player)
 {
-	msecnode_t *node;
 	boolean teeter = false;
 	boolean roverfloor; // solid 3d floors?
-	boolean checkedforteeter = false;
 	const fixed_t tiptop = FixedMul(MAXSTEPMOVE, player->mo->scale); // Distance you have to be above the ground in order to teeter.
 
-	for (node = player->mo->touching_sectorlist; node; node = node->m_thinglist_next)
-	{
-		// Ledge teetering. Check if any nearby sectors are low enough from your current one.
-		checkedforteeter = true;
-		roverfloor = false;
-		if (node->m_sector->ffloors)
-		{
-			ffloor_t *rover;
-			for (rover = node->m_sector->ffloors; rover; rover = rover->next)
-			{
-				if (!(rover->flags & FF_EXISTS)) continue;
-
-				if (P_CheckSolidLava(player->mo, rover))
-					;
-				else if (!(rover->flags & FF_BLOCKPLAYER || rover->flags & FF_QUICKSAND))
-					continue; // intangible 3d floor
-
-				if (player->mo->eflags & MFE_VERTICALFLIP)
-				{
-					if (*rover->bottomheight > node->m_sector->ceilingheight) // Above the ceiling
-						continue;
-
-					if (*rover->bottomheight > player->mo->z + player->mo->height + tiptop
-						|| (*rover->topheight < player->mo->z
-						&& player->mo->z + player->mo->height < node->m_sector->ceilingheight - tiptop))
-					{
-						teeter = true;
-						roverfloor = true;
-					}
-					else
-					{
-						teeter = false;
-						roverfloor = true;
-						break;
-					}
-				}
-				else
-				{
-					if (*rover->topheight < node->m_sector->floorheight) // Below the floor
-						continue;
-
-					if (*rover->topheight < player->mo->z - tiptop
-						|| (*rover->bottomheight > player->mo->z + player->mo->height
-						&& player->mo->z > node->m_sector->floorheight + tiptop))
-					{
-						teeter = true;
-						roverfloor = true;
-					}
-					else
-					{
-						teeter = false;
-						roverfloor = true;
-						break;
-					}
-				}
-			}
-		}
-
-		if (!teeter && !roverfloor)
-		{
-			if (player->mo->eflags & MFE_VERTICALFLIP)
-			{
-				if (node->m_sector->ceilingheight > player->mo->z + player->mo->height + tiptop)
-					teeter = true;
-			}
-			else
-			{
-				if (node->m_sector->floorheight < player->mo->z - tiptop)
-					teeter = true;
-			}
-		}
-	}
-
-	if (checkedforteeter && !teeter) // Backup code
 	{
 		subsector_t *subsec[4]; // changed abcd into array instead
 		UINT8 i;
@@ -3147,7 +3071,7 @@ teeterdone:
 		if ((player->mo->state == &states[S_PLAY_STND] || player->mo->state == &states[S_PLAY_TAP1] || player->mo->state == &states[S_PLAY_TAP2] || player->mo->state == &states[S_PLAY_SUPERSTAND]))
 			P_SetPlayerMobjState(player->mo, S_PLAY_TEETER1);
 	}
-	else if (checkedforteeter && (player->mo->state == &states[S_PLAY_TEETER1] || player->mo->state == &states[S_PLAY_TEETER2] || player->mo->state == &states[S_PLAY_SUPERTEETER]))
+	else if ((player->mo->state == &states[S_PLAY_TEETER1] || player->mo->state == &states[S_PLAY_TEETER2] || player->mo->state == &states[S_PLAY_SUPERTEETER]))
 		P_SetPlayerMobjState(player->mo, S_PLAY_STND);
 }
 
@@ -7037,7 +6961,7 @@ static void P_MovePlayer(player_t *player)
 		player->mo->y += player->mo->momy;
 		P_SetThingPosition(player->mo);
 
-		for (node = player->mo->touching_sectorlist; node; node = node->m_thinglist_next)
+		for (node = player->mo->touching_sectorlist; node; node = node->m_sectorlist_next)
 		{
 			if (!node->m_sector)
 				break;