Ambiguity in Earth's "Tilt" It’s well-known that the axial tilt of the Earth (with respect to the ecliptic) is about 23.4 degrees. However, two angles is needed to specify the orientation of any rigid body, so it’s unclear to me exactly how Earth is oriented with respect to its orbital plane.

To be more precise, consider the plane that satisfies these two conditions:
1. Perpendicular to the ecliptic plane
2. Contains the line connecting the Sun and the Earth when Earth is at perihelion (Jan. 2).

(Basically, this plane cuts through the orbit of the Earth such that for half the year Earth is one side and for the other half Earth is on the other side.)

Questions:
1. When the Earth is at perihelion, what is the angle between its rotational axis and the plane described above? I would guess this angle is very small from the images of Earth’s orbit. Is there a technical term for this angle?
2. Does the direction of the axis remain approximately the same throughout its entire orbit?

Please ignore precession for this question.
The fact that the perihelion is close to the the winter solstice is purely coincidental. The perihelion might occur in any season. It wanders around in a somewhat chaotic fashion because of the gravitational influence of the other planets. At the moment the perihelion moves a day every 58 years.

Let A = Earths axis, and E = the line perpendicular to the orbital plane. The angle between A and E is called the axial tilt. Let S = the line between the centers of the Earth and the Sun. Let P = the plane containing S and E. 

Q1 is (maybe) asking what is the angle between A and P on Jan 2. It will be about 23.4 times sine(2*pi*12/365).  Because there are 12 days from Dec 21 to Jan 2.

Q2. yes.