Remove duplicates in custom IComparable class

I have a table that has combo pairs identifiers, and I use that to go through CSV files looking for matches. I'm trapping the unidentified pairs in a List, and sending them to an output box for later addition. I would like the output to only have single occurrences of unique pairs. The class is declared as follows:

    public class Unmatched:IComparable<Unmatched>
{
    public string first_code { get; set; }
    public string second_code { get; set; }

    public int CompareTo(Unmatched other)
    {

        if (this.first_code == other.first_code)
        {
            return this.second_code.CompareTo(other.second_code);
        }

        return other.first_code.CompareTo(this.first_code);
    }
 }

One note on the above code: This returns it in reverse alphabetical order, to get it in alphabetical order use this line:

return this.first_code.CompareTo(other.first_code);

Here is the code that adds it. This is directly after the comparison against the datatable elements

unmatched.Add(new Unmatched() 
{ first_code = fields[clients[global_index].first_match_column]
, second_code = fields[clients[global_index].second_match_column] });

I would like to remove all pairs from the list where both first code and second code are equal, i.e.;

PTC,138A
PTC,138A
PTC,138A
MA9,5A
MA9,5A
MA9,5A
MA63,138A
MA63,138A
MA59,87BM
MA59,87BM

Should become:

PTC, 138A
MA9, 5A
MA63, 138A
MA59, 87BM

I have tried adding my own Equate and GetHashCode as outlined here: http://www.morgantechspace.com/2014/01/Use-of-Distinct-with-Custom-Class-objects-in-C-Sharp.html

The SE links I have tried are here:

How would I distinct my list of key/value pairs

Get list of distinct values in List<T> in c#

Get a list of distinct values in List

All of them return a list that still has all the pairs. Here is the current code (Yes, I know there are two distinct lines, neither appears to be working) that outputs the list:

    parser.Close();
    List<Unmatched> noDupes = unmatched.Distinct().ToList();
    noDupes.Sort();
    noDupes.Select(x => x.first_code).Distinct();

    foreach (var pair in noDupes)
    {
        txtUnmatchedList.AppendText(pair.first_code + "," + pair.second_code + Environment.NewLine);
    }

Here is the Equate/Hash code as requested:

public bool Equals(Unmatched notmatched)
{

    //Check whether the compared object is null.  
    if (Object.ReferenceEquals(notmatched, null)) return false;

    //Check whether the compared object references the same data.  
    if (Object.ReferenceEquals(this, notmatched)) return true;

    //Check whether the UserDetails' properties are equal.  
    return first_code.Equals(notmatched.first_code) && second_code.Equals(notmatched.second_code);
}

// If Equals() returns true for a pair of objects   
// then GetHashCode() must return the same value for these objects.  

public override int GetHashCode()
{

    //Get hash code for the UserName field if it is not null.  
    int hashfirst_code = first_code == null ? 0 : first_code.GetHashCode();

    //Get hash code for the City field.  
    int hashsecond_code = second_code.GetHashCode();

    //Calculate the hash code for the GPOPolicy.  
    return hashfirst_code ^ hashsecond_code;
}

I have also looked at a couple of answers that are using queries and Tuples, which I honestly don't understand. Can someone point me to a source or answer that will explain the how (And why) of getting distinct pairs out of a custom list?

(Side question-Can you declare a class as both IComparable and IEquatable?)