Of course, you can always punch with a sword's pommel or guard, or add a spear head to the other end of a pole weapon, but I'm talking about weapons that are SWUNG to attack with both ends. Something like this:
While a bit strange, double weapons are not uncommon in fiction. They are usually associated with characters that are both agile and strong. When I wrote my old Adryon setting, I gave the weapon picture above as a cultural weapon to the Bashti, my cat-people.
In D&D, 3rd edition had a whole class of double weapons. Here is how they work (from the SRD):
Dire flails, dwarven urgroshes, gnome hooked hammers, orc double axes, quarterstaffs, and two-bladed swords are double weapons. A character can fight with both ends of a double weapon as if fighting with two weapons, but he or she incurs all the normal attack penalties associated with two-weapon combat, just as though the character were wielding a one-handed weapon and a light weapon.
The character can also choose to use a double weapon two handed, attacking with only one end of it. A creature wielding a double weapon in one hand can’t use it as a double weapon—only one end of the weapon can be used in any given round.
That is a very good treatment of double weapons, since the traits we usually associate with "fighting with two weapons" (TWF) are them same we associate with "double weapons": dexterity, speed (more attacks), improved defense, less damage per attack.
It is not impossible to use the 3.5e rules in 5e almost as written. Just apply 5e's two-weapon fighting limitations to double weapons. You can attack with one end with a bonus action but you don’t add your ability modifier to the damage of the bonus action unless you have the fighting style that says you can.
Damage should be limited to 1d6/1d6 unless you can dual-wield weapons that aren't light (with the Dual Wielder feat, for example) - then you can get a heavier weapon that deals 1d8/1d8 damage, which is basically the same as wielding two rapiers.
Even better, you can keep the quarterstaff damage as written and give 1d4 damage to the other end so that characters with multiple attacks get a boost.
This is how a "double" weapon properties would look like in 5e:
Double. When you take the Attack action and attack with a double weapon that you’re holding in both hands, you can use a bonus action to attack with the other end of the weapon, causing 1d4 damage. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
If used this way, double weapons seem a bit better than TWF - mostly because of how they interact with things such as multiple attacks, magic item attunement, object interaction, spells such as Magic Weapon and Shillelagh, etc. - but they cannot be thrown or used with finesse, which balances things a bit.
Since we're doing feats, we might probably distinguish double weapon fighting from two weapon fighting with a feat. Here is my take:
Double Weapon Master
You have mastered techniques that allow you to fight effectivly with both ends of a single weapon. You can treat the quarterstaff as a double weapon. In addition, while wielding a double weapon with both hands, you gain the following benefits:
As you can see, the second bullet point is from my last post on two-weapon fighting.
The feat still looks underwhelming when compared to Polearm Master (one of the best feats in the PHB, IMO), but might be used with the exotic double weapons, in addition to the increased accuracy and AC. It is better than PAM to deal damage consistently on your turn, but worse when trying to "control the battlefield" against incoming enemies.
Better for dueling, worse for war. Makes plenty of sense for double weapons.