Equals vs GetHashCode when comparing objects

Should we override both Equals and GetHashCode properties when implementing a custom class instances comparison?

In the following code I have a collection of classes. The class A is compared by the ID, the class B - by Code.

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

namespace ConsoleApplication2
{
    class Program
    {
        static void Main(string[] args)
        {
            List<I> bars = new List<I>();
            bars.Add(new A() { Id = 1, Code = "one A" });
            bars.Add(new B() { Id = 1, Code = "one B" });
            bars.Add(new A() { Id = 1, Code = "one A+" });
            bars.Add(new B() { Id = 1, Code = "one B" }); // Code = "one B+"

            var distictBars = bars.Distinct();

            foreach (var item in distictBars)
            {
                Debug.WriteLine(item.Code);
            }
        }
    }

    interface I
    {
        string Code { get; set; }
    }

    class A : I, IEquatable<A>
    {
        public int Id { get; set; }
        public string Code { get; set; }

        public bool Equals(A other)
        {
            // this is the ??? comparison
            return this.Id == other.Id;
            //return this.Code == other.Code;
        }

        public override bool Equals(object obj)
        {
            if (obj is A)
                return this.Equals(obj as A);
            else
                return object.ReferenceEquals(this, obj);
        }

        public override int GetHashCode()
        {
            // this is the wanted comparison
            return this.Id;
        }
    }

    class B : I, IEquatable<B>
    {
        public int Id { get; set; }
        public string Code { get; set; }

        public bool Equals(B other)
        {
            // this is the ??? comparison