The peace offering is a unique sacrifice that fosters a sense of connection and joy, as the animal is shared among the altar, the person bringing the offering, and the priests. Handing a specific portion to the priest is far more than a technical procedure; it carries deep significance regarding the relationship between the physical and the spiritual, and between an individual and God's servants.
The specified portion is the upper part of the animal's hind legs, specifically the thick, fleshy section between the lower and middle joints [רש״י, ביאור יש״ר, פרדס יוסף, ברכת אשר]. Because this leg muscle is used for running, it represents a premium cut of meat that was historically reserved for highly honored guests [רד צ הופמן]. By taking this choice piece from their own share and giving it to the priest, the owner acts like someone offering a gift to a king's servant as a way to honor the king Himself [ספורנו]. Interestingly, while ancient nations would burn this specific cut on their altars to honor their idols, the Torah repurposed it, directing it to be eaten by the priests rather than burned [אם למקרא].
In practice, giving the right thigh of a holy offering parallels the commandment to give the foreleg of regular, non-sacred animals to the priest. This requirement broadly applies to all types of sacrifices, including the peace offerings brought by a Nazarite [תורה תמימה].
Commentators note a fundamental difference between the thigh and the breast of the animal, which is also given to the priest. In terms of original ownership, the breast is viewed as a portion from God's table that He grants to the priests, whereas the thigh is taken directly from the owner's personal share [העמק דבר]. This distinction is also reflected in how they are handled in the Temple. The breast is moved forward and backward, mimicking its natural position in the animal, while the thigh is lifted up and down, resembling the motion of a leg during walking [רלב״ג].
There is a deep conceptual divide between these two parts. The breast symbolizes thought and will, areas that naturally align with the priest's spiritual focus. The thigh, however, represents physical strength, practical progress, and material wealth. When a person lifts the thigh as an offering, they declare that their material success and physical abilities draw their life force from God's teachings, which the priests represent. Through this act, they dedicate their physical power to support these spiritual values [רש ר הירש]. According to mystical teachings, lifting up this physical portion elevates the material branch back to its spiritual root [רקנאטי]. Ultimately, by giving away the right thigh, a person connects their material abundance directly to the source of all blessing.