A Monster Card equipped with this card increases its ATK and DEF by 500 points. When this card is sent to the Graveyard, place it on top of your Deck.

If i equipped this card to a Gladiator Beast, and after i attack, i send th GB back to the deck, to get another, does Sword of Deep-Seated go to the top of the deck first or after GB effect?

Meaning, Sword of Deep-Seated first, no longer on top cos i need to shuffle deck.

Ooh, that is a smidge tricky, but the way I see it is this.

During the end phase, you activate the effect of the Gladiator Beast, returning to the deck as the cost.

Due to game mechanics, Sword of Deep-Seated will be sent to the graveyard immediately. This will trigger the effect of the sword, but we are still resolving the effect of the Gladiator Beast, so it won't activate.

You resolve the effect of the Gladiator Beast card accordingly, special summoning another Gladiator Beast and shuffling your deck.

Now, since Sword of Deep-Seated's effect is not an optional trigger effect, it will now activate, and not barring any negation, it will return to the top of your deck.

That's how I see this playing out. Anybody else see care to comment?

I think you did a pretty good job of explaining the gist of it Dillie-O. SoDS waits its turn to start a new chain after the current one resolves.