C#-Code

using static System.Console; //VS 2019 .Net 5.0.2

System.Text.StringBuilder sb = new System.Text.StringBuilder();

int[] tage_im_monat = new int[] { 31, 29, 31, 30, 31, 30, 31, 31, 30, 31, 30, 30 };
int start = 000, end = 10000;

static string sx(string s, int x) => s.Length == x ? s : sx("0" + s, x);

int count = 0;
for (var i = start; i < end; i++)
    for (var j = 0; j < tage_im_monat.Length; j++)    {
        string sm = sx((j + 1).ToString(), 2);
        for (var k = 0; k < tage_im_monat[j]; k++)    {
            string st = sx((k + 1).ToString(), 2);
            string temp = st + sm + sx(i.ToString(), 4);
            bool bo_palindrom = true;
            for (var l = 0; l < 4; l++)
                if (temp[l] != temp[temp.Length - 1 - l])
                    bo_palindrom = false;
            if (bo_palindrom)
            {
                WriteLine($"{++count,3}: {temp}    {int.Parse(temp.Substring(4)),4}");               
                sb.Append($"{temp}, ");
            }
        }        
    }
//System.IO.File.WriteAllText("erg_palindrom.txt", sb.ToString());
ReadKey();