Short answer. Yes, but only in the proper cases. The rule allows a third-party defendant to allege defenses, counterclaims, or cross-claims, and specifically adds that in proper cases, he may also assert a counterclaim against the original plaintiff in respect of the latter's claim against the third-party plaintiff.
What the law says
In proper cases, he [or she] may also assert a counterclaim against the original plaintiff in respect of the latter's claim against the third-party plaintiff
Rule 6, Section 13 — Answer to third (fourth, etc.). Read the full provision →
What a third-party defendant can raise in the answer generally
A third-party defendant is entitled to allege in the answer his or her defenses, counterclaims or cross-claims, and that includes such defenses that the third-party plaintiff may have against the original plaintiff's claim. That means you are not limited only to defending against whatever claim the third-party plaintiff — the specific party who actually brought you into the case — has personally made against you at all.
The specific right to counterclaim against the original plaintiff
Beyond raising the third-party plaintiff's own defenses, the rule separately and expressly allows something more direct: in proper cases, you may assert your own counterclaim against the original plaintiff, tied to that plaintiff's claim against the third-party plaintiff. This is a distinct right from defending yourself against the person who impleaded you — it lets you go on the offensive directly against the plaintiff whose original claim ultimately triggered your being brought into the case in the first place, rather than confining you to a purely defensive role throughout the proceeding.
Why the phrase 'in proper cases' matters
This particular right is not entirely unconditional. The rule limits it to proper cases, signaling that whether a counterclaim against the original plaintiff is appropriate depends on the specific circumstances — how your position connects to the original plaintiff's claim against the third-party plaintiff, and whether the counterclaim genuinely relates closely to that underlying claim, rather than being an unrelated matter you happen to also have against that same plaintiff.
The connection to the original claim is what anchors this right
Notice carefully that the counterclaim the rule actually contemplates is specifically in respect of the latter's claim against the third-party plaintiff — meaning it has to relate directly to the very same claim that put you into the case in the first place, not any separate grievance you might otherwise have against the original plaintiff on some entirely unconnected matter or transaction. This connection requirement binds the third-party defendant specifically — it does not open the door to litigating every dispute you might have with the plaintiff in the same proceeding.