BalancedLatinSquares1.cs

public List<List<String>> BalancedLatinSquares(int n)
{
    var result = new List<List<String>>() { };
    for (int i = 0; i < n; i++)
    {
        var row = new List<String>();
        for (int j = 0; j < n; j++)
        {
            var cell = ((j % 2 == 1 ? j / 2 + 1 : n - j / 2) + i) % n;
            cell++; // start counting from 1
            row.Add(cell.ToString());
        }
        result.Add(row);
    }
    if (n % 2 == 1)
    {
        var reversedResult = result.Select(x => x.AsQueryable().Reverse().ToList()).ToList();                
        result.AddRange(reversedResult);
    }
    return result;
}

BalancedLatinSquares2.cs

public List<List<String>> BalancedLatinSquares(int n)
{
    var result = Enumerable.Range(0, n)
        .Select(i =>
                Enumerable.Range(0, n).Select(j => ((((j % 2 == 1 ? j / 2 + 1 : n - j / 2) + i) % n)+1).ToString()).ToList()
            )
        .ToList();     
    
    if (n % 2 == 1)
    {
        var reversedResult = result.Select(x => x.AsQueryable().Reverse().ToList()).ToList();                
        result.AddRange(reversedResult);
        return result;
}

BalancedLatinSquares3.cs

public static IEnumerable<IEnumerable<int>> GenerateBalancedLatinSquares(int n)
{
    bool isEven (int i) => i % 2 == 0;        
    int calulateCell(int j, int i) => ((isEven(j) ? n - j / 2 : j / 2 + 1) + i) % n + 1;
    
    var result = Enumerable
        .Range(0, n)
        .Select(i =>
            Enumerable
                .Range(0, n)
                .Select(j =>calulateCell(j,i))
        );     
    
    if (n % 2 == 1)
    {
        var reversedResult = result.Select(x => x.Reverse());                
        result = result.Concat(reversedResult);
    }        return result;
}     

FinalBalancedLatinSquares.cs

public static class BalancedLatinSquare
{               
    /**
     * Orders the given sequence in from of a balanced latin square 
     */
    public static IEnumerable<IEnumerable<T>> Arrange<T>(IEnumerable<T> items)
    {
        var latinSquare = GenerateBalancedLatinSquares(items.Count());
        return latinSquare.Select(row => row.Select(cellIndex => items.ElementAt(cellIndex-1)).ToList()).ToList();
    }    
    
    /**
     * Generates a balanced latin square for the numbers 1 to n
     */
    public static IEnumerable<IEnumerable<int>> GenerateBalancedLatinSquares(int n)
    {
        bool isEven (int i) => i % 2 == 0;        
        int calculateCell(int j, int i) => ((isEven(j) ? n - j / 2 : j / 2 + 1) + i) % n + 1;
        
        var result = Enumerable
            .Range(0, n)
            .Select(i =>
                Enumerable
                    .Range(0, n)
                    .Select(j =>calulateCell(j,i))
            );     
        
        if (n % 2 == 1)
        {
            var reversedResult = result.Select(x => x.Reverse());                
            result = result.Concat(reversedResult);
        }        return result;
    }                
}