Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?
1. **Determine the height of the pyramid:**
   Each edge of the pyramid is 12 cm. The base of the pyramid is a square with side length \( a \). Since all edges are equal, the slant height \( s \) of the pyramid can be found using the Pythagorean theorem in the right triangle formed by the slant height, half the base, and the height \( h \) of the pyramid.

   \[
   s^2 = \left(\frac{a}{2}\right)^2 + h^2
   \]

   Given \( s = 12 \) cm and \( a = 12 \) cm, we have:

   \[
   12^2 = \left(\frac{12}{2}\right)^2 + h^2
   \]

   \[
   144 = 36 + h^2
   \]

   \[
   h^2 = 108
   \]

   \[
   h = \sqrt{108} = 6\sqrt{3} \text{ cm}
   \]

2. **Determine the height of the intersection:**
   The height of the intersection is the distance from the apex of one pyramid to the base of the other pyramid. Since the apex of each pyramid is at the center of the base of the other pyramid, the height of the intersection is half the height of one pyramid:

   \[
   \text{Height of intersection} = \frac{6\sqrt{3}}{2} = 3\sqrt{3} \text{ cm}
   \]

3. **Determine the base area of the intersection:**
   The intersection forms an octahedron-like figure. The base of each of the two congruent pyramids forming the intersection is a square with side length equal to half the side length of the original pyramid's base:

   \[
   \text{Side length of base} = \frac{12}{2} = 6 \text{ cm}
   \]

   \[
   \text{Base area} = 6^2 = 36 \text{ cm}^2
   \]

4. **Determine the volume of each pyramid in the intersection:**
   Each pyramid in the intersection has a base area of 36 cm\(^2\) and a height of 3 cm:

   \[
   \text{Volume of one pyramid} = \frac{1}{3} \times \text{Base area} \times \text{Height}
   \]

   \[
   \text{Volume of one pyramid} = \frac{1}{3} \times 36 \times 3 = 36 \text{ cm}^3
   \]

5. **Determine the total volume of the intersection:**
   Since the intersection consists of two congruent pyramids:

   \[
   \text{Total volume} = 2 \times 36 = 72 \text{ cm}^3
   \]

The final answer is \(\boxed{72} \text{ cm}^3\).