Reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected.^{[1]} Methods that satisfy reversal symmetry include Borda count, ranked pairs, Kemeny–Young method, and Schulze method. Methods that fail include Bucklin voting, instantrunoff voting and Condorcet methods that fail the Condorcet loser criterion such as Minimax.
For cardinal voting systems which can be meaningfully reversed, approval voting and range voting satisfy the criterion.
Main article: Instantrunoff voting 
Consider a preferential system where 11 voters express their preferences as:
With the Borda count A would get 23 points (5×3+4×1+2×2), B would get 24 points, and C would get 19 points, so B would be elected. In instantrunoff, C would be eliminated in the first round and A would be elected in the second round by 7 votes to 4.
Now reversing the preferences:
With the Borda count A would get 21 points (5×1+4×3+2×2), B would get 20 points, and C would get 25 points, so this time C would be elected. In instantrunoff, B would be eliminated in the first round and A would as before be elected in the second round, this time by 6 votes to 5.
Main article: Majority Judgment 
This example shows that Majority Judgment violates the Reversal symmetry criterion. Assume two candidates A and B and 2 voters with the following ratings:
Candidates/ # of voters 
A  B 

1  Good  Fair 
1  Poor  Fair 
Now, the winners are determined for the normal and the reversed ballots.
In the following the Majority Judgment winner for the normal ballots is determined.
Candidates/ # of voters 
A  B 

1  Good  Fair 
1  Poor  Fair 
The sorted ratings would be as follows:
Candidate 
 
A 
 
B 
 

Result: The median of A is between "Good" and "Poor" and thus is rounded down to "Poor". The median of B is "Fair". Thus, B is elected Majority Judgment winner.
In the following the Majority Judgment winner for the reversed ballots is determined. For reversing, the higher ratings are considered to be mirrorinverted to the lower ratings ("Good" is exchanged with "Poor", "Fair" stays as is).
Candidates/ # of voters 
A  B 

1  Poor  Fair 
1  Good  Fair 
The sorted ratings would be as follows:
Candidate 
 
A 
 
B 
 

Result: Still, the median of A is between "Good" and "Poor" and thus is rounded down to "Poor". The median of B is "Fair". Thus, B is elected Majority Judgment winner for the reversed ballots.
B is the Majority Judgment winner using the normal ballots and also using the ballots with reversed ratings. Thus, Majority Judgment fails the Reversal symmetry criterion.
However, note that using another rounding method could prevent the failure to Reversal symmetry. Also, note that this situation is unlikely to arise in practical elections with many voters because it involves a "tie" of sorts  some candidate (A in this case) gets exactly the same number of votes above and below a certain value ("fair" in this case).
Main article: Minimax Condorcet 
This example shows that the Minimax method violates the Reversal symmetry criterion. Assume four candidates A, B, C and D with 14 voters with the following preferences:
# of voters  Preferences 

4  A > B > D > C 
4  B > C > A > D 
2  C > D > A > B 
1  D > A > B > C 
1  D > B > C > A 
2  D > C > A > B 
Since all preferences are strict rankings (no equals are present), all three Minimax methods (winning votes, margins and pairwise opposite) elect the same winners.
Now, the winners are determined for the normal and the reversed order.
In the following the Minimax winner for the ballots in normal order is determined.
# of voters  Preferences 

4  A > B > D > C 
4  B > C > A > D 
2  C > D > A > B 
1  D > A > B > C 
1  D > B > C > A 
2  D > C > A > B 
The results would be tabulated as follows:
X  
A  B  C  D  
Y  A  [X] 5 [Y] 9 
[X] 9 [Y] 5 
[X] 6 [Y] 8  
B  [X] 9 [Y] 5 
[X] 4 [Y] 10 
[X] 6 [Y] 8  
C  [X] 5 [Y] 9 
[X] 10 [Y] 4 
[X] 8 [Y] 6  
D  [X] 8 [Y] 6 
[X] 8 [Y] 6 
[X] 6 [Y] 8 

Pairwise election results (wontiedlost):  201  201  102  102  
worst pairwise defeat (winning votes):  9  9  10  8  
worst pairwise defeat (margins):  4  4  6  2  
worst pairwise opposition:  9  9  10  8 
Result: The candidates A, B, and C form a cycle with clear defeats. D benefits from that since its two losses are relatively close and therefore D's biggest defeat is the closest of all candidates. Thus, D is elected Minimax winner.
In the following the Minimax winner for the ballots in reversed order is determined.
# of voters  Preferences 

4  C > D > B > A 
4  D > A > C > B 
2  B > A > D > C 
1  C > B > A > D 
1  A > C > B > D 
2  B > A > C > D 
The results would be tabulated as follows:
X  
A  B  C  D  
Y  A  [X] 9 [Y] 5 
[X] 5 [Y] 9 
[X] 8 [Y] 6  
B  [X] 5 [Y] 9 
[X] 10 [Y] 4 
[X] 8 [Y] 6  
C  [X] 9 [Y] 5 
[X] 4 [Y] 10 
[X] 6 [Y] 8  
D  [X] 6 [Y] 8 
[X] 6 [Y] 8 
[X] 8 [Y] 6 

Pairwise election results (wontiedlost):  102  102  201  201  
worst pairwise defeat (winning votes):  9  10  9  8  
worst pairwise defeat (margins):  4  6  4  2  
worst pairwise opposition:  9  10  9  8 
Result: Still, the candidates A, B, and C form a cycle with clear defeats and D benefits from that. Therefore D's biggest defeat is the closest of all candidates. Thus, D is elected Minimax winner.
D is the Minimax winner using the normal preference order and also using the ballots with reversed preference orders. Thus, Minimax fails the Reversal symmetry criterion.
Main article: Plurality voting system 
This example shows that Plurality voting violates the Reversal symmetry criterion. Assume three candidates A, B and C and 4 voters with the following preferences:
# of voters  Preferences 

1  A > B > C 
1  C > B > A 
1  B > A > C 
1  C > A > B 
Note that reversing all the ballots, leads to the same set of ballots, since the reversed preference order of the first voter resembles the preference order of the second, and similarly with the third and fourth.
In the following the Plurality winner is determined. Plurality ballots only contain the single favorite:
# of voters  Favorite 

1  A 
1  B 
2  C 
Result: The candidates A and B receive 1 vote each, candidate C receives a plurality of 2 votes (50%). Thus, C is elected Plurality winner.
C is the Plurality winner using the normal ballots and also using the reversed ballot. Thus, Plurality fails the Reversal symmetry criterion.
Note, that every voting system that satisfies the Reversal symmetry criterion, would have to lead to a tie in this example (as in every example in which the set of reversed ballots is the same as the set of normal ballots).
Main article: STAR voting 
This example shows STAR violates the reversal symmetry criterion. In a score ballot, reversed score is calculated as the maximum possible score minus the normal score.
Given a 3candidate election between candidates A, B, and C:
Candidate  Ballots  

A  5  5  2  2  2 
B  0  0  3  3  3 
C  1  1  2  0  4 
The results are tabulated below:
Candidate  Total Score  Preferred vs  

A  B  C  
A  16    2  3 
B  9  3    2 
C  8  1  3   
Result: In the election, candidates A and B have the highest scores, and advance to the runoff round. B wins being preferred over A 3 votes to 2.
Reversing the ballots by subtracting each score from 5 (the maximum score in STAR) gives the following:
Candidate  Ballots  

A  0  0  3  3  3 
B  5  5  2  2  2 
C  4  4  3  5  1 
The results are tabulated below:
Candidate  Total Score  Preferred vs  

A  B  C  
A  9    3  1 
B  16  2    3 
C  17  3  2   
Result: In the reversed ballots, B and C have the highest total score, and B wins being preferred to C 3 votes to 2.
The examples above show that the same candidate, B, wins in both the normal and reversed election. Thus, STAR violates the reversal symmetry criterion.