The earth's axis is titled 23.4 degrees with respect to the ecliptic. Since this tilt is responsible for the seasons, it's clear that the tilt is in the direction away from the sun on the shortest day of year in the winter (on the northern hemisphere). That uniquely defines the tilt direction for me. How do we define the tilt direction for the other planets, though? What is the proper way of determining in what direction the tilt of a planet occurs?
A bit too simply (but not too much), it's the angle between the planet's rotational angular velocity vector and the planet's orbital angular momentum vector.
Where it gets tricky is dealing with little wobbles and such. Planets don't quite orbit in a plane because of gravitational interactions amongst the planets. This means the instantaneous (osculating) orbital angular momentum vector isn't quite constant, both in magnitude and direction. There are a number of different mean orbital elements that smooth out most of those tiny wobbles. One of those mean orbital elements sets (I'm not sure which one exactly) is used to determine the mean orbital angular momentum vector.
Planets don't quite rotate nice and smoothly because planets aren't perfect spheres and aren't rigid bodies. This gives the Moon, the Sun, and other planets a handle by which they can exert tiny little torques in the planet in question. The response of the planet to these torques is somewhat arbitrarily divided into two categories based on frequency. Very slow responses are called precession; faster responses, nutation. The non spherical nature of a planet means the planet undergoes a small torque-free nutation as well as the torque-induced precession, nutation, and polar motion. In addition to these, there are a number of terms (all small) that don't yet have a very good model behind them. This oddball terms, along with the torque-free nutation, are collectively called polar motion. The motion is fairly smooth if one ignores nutation and polar motion, and the effects are all small.
Precession, while very slow, can be quite large in magnitude. The rotational angular velocity vector that is used in determining the axial tilt incorporates precession but smooths out nutation and polar motion.
The tilt doesn't depend at all on where one is on the planet, and whether or not it is to the left or right is irrelevant.
The angle (excluding the wobbles and longer term effects as David mentioned) doesn't vary depending on where you measure it. See this pic from NOAA for a simple perspective:
I think I found the answer to my question. The rotational axis of each planet is identical with the vector pointing from the planet's south pole to its north pole. Therefore, if I find a table that defines positions of north poles of the planets, I will also find the unique vector pointing in the direction of the planets' tilts. This page http://en.wikipedia.org/wiki/Poles_of_astronomical_bodies shows the positions of the north poles of major bodies in the solar system (as well as their tilt angles) projected onto the celestial sphere. The coordinates are given relative to Earth's celestial equator and the vernal equinox as they existed at the J2000 epoch. These coordinates uniquely define the tilt of each of the planets and other relatively stable bodies within the solar system.