Short answer. Yes. Where a judgment rests on the obligation of one person as security for another and the surety pays the amount or any part of it, the rule states plainly that he may compel repayment from the principal — you are entitled to be reimbursed by the debtor you stood surety for.
What the law says
when a judgment is upon an obligation of one of them, as security for another, and the surety pays the amount, or any part thereof, either by sale of his property or before sale, he may compel repayment from the principal
Rule 39, Section 35 — Right to contribution or reimbursement. Read the full provision →
The rule speaks to your exact situation
The provision specifically addresses a judgment upon an obligation of one of them, as security for another — precisely the surety-and-principal relationship you now describe. Once the surety pays that judgment, the rule gives an express right of recovery: the surety may compel repayment from the principal. It is not framed as a discretionary favor the principal might grant, but as an actual right the surety can enforce directly, regardless of whether the principal is willing to cooperate.
It does not matter exactly how you ended up paying
The right to seek repayment applies whether the surety pays either by sale of his property or before sale, and whether the payment covers the full amount or only part of it. Whether your property was sold to satisfy the judgment or you paid before that happened, and whether you paid everything or only a portion, the right to be repaid by the principal follows the same way.
How this differs from the contribution right among co-debtors
The same provision separately covers a different scenario — several persons liable on one judgment, where one of them pays more than his own proportionate share and may compel contribution from the others. That rule involves co-debtors sharing a burden. Your situation is different: you did not owe a proportionate share of a shared debt, you paid as surety for someone else's obligation, so the remedy available to you is full repayment from the principal, not the partial, proportionate contribution co-debtors would owe each other.
What this means practically for you
Having paid the judgment as surety puts you in the position the rule addresses directly — you are fully entitled to recover what you paid from the principal debtor. The rule frames this as your right to compel repayment, giving you a clear, enforceable legal basis to pursue the principal debtor for the full amount you actually ended up paying on that person's own underlying obligation to the creditor.