Here's my take on all of this: I believe that there is a fourth goddess, and there was originally intended to be a fourth piece of the Triforce.
Numbers
There are four Links in Four Swords, and a lot of "bad" things come in threes and their multiples (3 strikes on a boss, 3-6 temples, etc.). Additionally, there were four silent realms in SS, four providences in TP, and so on. There are also four main races in Hyrule in most games.
Symbols
The upside-down triangle or upside-down Triforce appears in the games almost as frequently as the symbol of the Triforce itself: OoT Link's Shield, the symbol on the Shadow Medallion, etc. Additionally, Tetra's necklace and the markings on the back of Midna's hands.
History
Now, it's already been mentioned that numbers and symbols don't create enough of a basis to theorize that there's another piece of the Triforce, so I'll explain the remainder of my evidence more through a story explaining why there isn't actually a fourth piece of the Triforce, what happened to it/Hylia, and why folks remember Din, Faore, and Nayru, but not Hylia.
We know that in the beginning, the three known goddesses created the world. Din the earth, Nayru the law, Faore the life, and Hylia the spirit of Hyrule. Being that each goddess gave the land something representative of their personalities, Hylia's personality was very loving and affectionate. She became attached to the world they had created, and when the other three goddesses departed for the heavens and left their pieces of the Triforce behind to protect their sister (most likely from the dark powers of Demise or another demon god), Hylia remained behind on the earth. Then Demise rose to attack the more vulnerable goddess and her people, so Hylia used all of her power--the power that she would have used to create the centerpiece of the Triforce --to save her chosen race. Weak and dying, she retreated into a Silent Realm to heal before placing her spirit in the heart of a mortal. As this mortal, she finally was able to place the power of her Triforce into a corporeal object: The Master Sword. In Skyward Sword, after Link has already acquired all of the Sacred Flames and the Master Sword is said to have "attained it's final form", Zelda blesses it once more and it attains its "ultimate form". This is Hylia putting the power of her Triforce into the sword.
After SS but before MC, Hyrule was settled. However, there were those who were resentful of being forgotten by the goddess. The Sheikah people had fought and given most of their lives to protecting the goddess and her plans, but after Zelda died they had been utterly forgotten. A small group of Sheikah used the remnants of Ghirahim and Demise's magic scattered throughout the land to construct powerful magical artifacts designed to rival the Triforce. They came to be known as the Fused Shadows. As the Interlopers rained terror on those who had forgotten them, they heard of a Sacred Realm left open to the world. The goddess statue brought down from Skyloft still presumably held the Triforce, and it is most likely that the Temple of Time designed to hide and protect the Sacred Realm wasn't built yet. The Interlopers pursued it. According to Hylia's plans, a princess containing her Spirit and a boy containing the Hero's Spirit were born to stop the Interlopers, but they made their coup before the princess of the Hero were ready. The other goddesses had to intervene, and they banished the Interlopers to the same place they'd imprisoned Demise: the Twilight Realm. Hylia/Zelda, however, tried to defend them and keep them in Hyrule. Already afraid of their sister being corrupted by her repeated interaction with Demise, the other goddesses tossed Hylia and her power in the Master Sword into the Twilight.
Hence the heavily reduced power of the Four Sword in MC and FS, as well as the increasing power of the Master Sword in OoT and TP when the Twilight was connecting with Hyrule once again. Also explains the repeated appearance of purple and a fourth sacred statue/carving/symbol in various games (the statues with the mashed faces in WW, anyone?).
At any rate, that's my headcanon for the Tetraforce. It actually ends up explaining a lot of confusing things about the Zelda-verse, such as why the Master Sword isn't always mega-powerful, why Link has to go through ridiculous "spirit-building" trials to possess the entire Triforce, or why the Triforce splits apart all the time.
Numbers
There are four Links in Four Swords, and a lot of "bad" things come in threes and their multiples (3 strikes on a boss, 3-6 temples, etc.). Additionally, there were four silent realms in SS, four providences in TP, and so on. There are also four main races in Hyrule in most games.
Symbols
The upside-down triangle or upside-down Triforce appears in the games almost as frequently as the symbol of the Triforce itself: OoT Link's Shield, the symbol on the Shadow Medallion, etc. Additionally, Tetra's necklace and the markings on the back of Midna's hands.
History
Now, it's already been mentioned that numbers and symbols don't create enough of a basis to theorize that there's another piece of the Triforce, so I'll explain the remainder of my evidence more through a story explaining why there isn't actually a fourth piece of the Triforce, what happened to it/Hylia, and why folks remember Din, Faore, and Nayru, but not Hylia.
We know that in the beginning, the three known goddesses created the world. Din the earth, Nayru the law, Faore the life, and Hylia the spirit of Hyrule. Being that each goddess gave the land something representative of their personalities, Hylia's personality was very loving and affectionate. She became attached to the world they had created, and when the other three goddesses departed for the heavens and left their pieces of the Triforce behind to protect their sister (most likely from the dark powers of Demise or another demon god), Hylia remained behind on the earth. Then Demise rose to attack the more vulnerable goddess and her people, so Hylia used all of her power--the power that she would have used to create the centerpiece of the Triforce --to save her chosen race. Weak and dying, she retreated into a Silent Realm to heal before placing her spirit in the heart of a mortal. As this mortal, she finally was able to place the power of her Triforce into a corporeal object: The Master Sword. In Skyward Sword, after Link has already acquired all of the Sacred Flames and the Master Sword is said to have "attained it's final form", Zelda blesses it once more and it attains its "ultimate form". This is Hylia putting the power of her Triforce into the sword.
After SS but before MC, Hyrule was settled. However, there were those who were resentful of being forgotten by the goddess. The Sheikah people had fought and given most of their lives to protecting the goddess and her plans, but after Zelda died they had been utterly forgotten. A small group of Sheikah used the remnants of Ghirahim and Demise's magic scattered throughout the land to construct powerful magical artifacts designed to rival the Triforce. They came to be known as the Fused Shadows. As the Interlopers rained terror on those who had forgotten them, they heard of a Sacred Realm left open to the world. The goddess statue brought down from Skyloft still presumably held the Triforce, and it is most likely that the Temple of Time designed to hide and protect the Sacred Realm wasn't built yet. The Interlopers pursued it. According to Hylia's plans, a princess containing her Spirit and a boy containing the Hero's Spirit were born to stop the Interlopers, but they made their coup before the princess of the Hero were ready. The other goddesses had to intervene, and they banished the Interlopers to the same place they'd imprisoned Demise: the Twilight Realm. Hylia/Zelda, however, tried to defend them and keep them in Hyrule. Already afraid of their sister being corrupted by her repeated interaction with Demise, the other goddesses tossed Hylia and her power in the Master Sword into the Twilight.
Hence the heavily reduced power of the Four Sword in MC and FS, as well as the increasing power of the Master Sword in OoT and TP when the Twilight was connecting with Hyrule once again. Also explains the repeated appearance of purple and a fourth sacred statue/carving/symbol in various games (the statues with the mashed faces in WW, anyone?).
At any rate, that's my headcanon for the Tetraforce. It actually ends up explaining a lot of confusing things about the Zelda-verse, such as why the Master Sword isn't always mega-powerful, why Link has to go through ridiculous "spirit-building" trials to possess the entire Triforce, or why the Triforce splits apart all the time.