TileSearcher.cs

using System;
using System.Collections.Generic;
using System.Linq;

namespace TilePuzzle
{
    enum TileColors { Red, Blue, Yellow, Green }

    class Program
    {
        const int ColorCount = 4;
        static void Main(string[] args)
        {
            var Permutations = new List<byte[]>();

            for (byte a = 0; a < ColorCount; a++)
            for (byte b = 0; b < ColorCount; b++)
            for (byte c = 0; c < ColorCount; c++)
            for (byte d = 0; d < ColorCount; d++)
            {
                Permutations.Add(new byte[] { a, b, c, d});
            }

            Console.WriteLine($"All possible tiles: {Permutations.Count}");
            Console.WriteLine();
            Console.WriteLine("Removing rotational duplicates...");

            var Combinations = new List<byte[]>();

            foreach (var p in Permutations)
            {
                var rotations = new[] { Turn90(p), Turn180(p), Turn270(p) };
                var dup = rotations.FirstOrDefault(r => Combinations.Any(c => c[0] == r[0] && c[1] == r[1] && c[2] == r[2] && c[3] == r[3]));
                if (dup != null)
                {
                    Console.WriteLine($"  Removing {TileToString(p)} because it duplicates {TileToString(dup)}");
                    continue;
                }

                Combinations.Add(p);
            }

            Console.WriteLine();
            Console.WriteLine($"All combinations without duplicates:");
            foreach(var tile in Combinations)
            {
                Console.WriteLine($"  {TileToString(tile)}");
            }

            Console.WriteLine();
            Console.WriteLine($"All combinations (without duplicates): {Combinations.Count}");
        }

        static byte[] Rotate(byte[] original) => new byte[] { original[1], original[2], original[3], original[0] };
        static byte[] Turn90(byte[] original) => Rotate(original);
        static byte[] Turn180(byte[] original) => Rotate(Rotate(original));
        static byte[] Turn270(byte[] original) => Rotate(Rotate(Rotate(original)));

        static string TileToString(byte[] tile) => $"[{string.Join(", ", tile.Select(b => (TileColors)b))}]";
    }
}