Or even when you travel from one US coast to another, you are standing at a ~45 degree tilt relative to how you were standing before.
I had thoughts like this swirling in my head during one moonless night on the pacific coast, as the campfire died down and our very own milky way galaxy came into view.
Us, our planet, and our solar system...we all live in the "suburbs" of our milky way galaxy, in one of its spiral arms.
So, when we see the milky way in the sky we can gaze toward the shiny lights "downtown," near the center of our galaxy. We can also gaze at other suburbs of our galaxy, getting an edge-on view of the spiral arms.
 |
| Image adapted from William Castleman. |
The campfire was now completely gone, and as the earth continued to turn, the milky way moved higher and higher in the sky. Pretty soon it was straight overhead--the spiral arms stretched above my head toward the northern horizon and the galactic center at my feet toward the southern horizon.
"The milky way and I sure are aligned tonight," I thought. Pondering this a bit longer, I realized that I was laying down parallel to the plane of our galaxy.
The milky way was huge and cosmically more significant than me, but my body was still oriented in a particular direction with respect to it. Crazy! Had I been laying on the ground oriented west-to-east I would have been floating in space perpendicular instead of parallel to the plane of our galaxy. Think about it!
 |
| Me, floating parallel to the plane of the milky way galaxy. Not to scale. Actually this is the nearby Andromeda Galaxy. Adapted from UniverseToday. |
My sense of north-south-east-west was not exact, but hopefully you get the idea. I was lying roughly north-to-south...I was parallel to the plane of the galaxy...therefore the orbit of the earth and its north-south axis must also be parallel to the plane of the galaxy.
In order for this to be possible, we--us, our solar system--must be oriented perpendicular-ish to the plane of the galaxy. Something like this:
 |
| The solar system and the milky way, tilted. Note: SUPER not to scale. |
It turns out I'm not totally bonkers, and the solar system is "tilted" with respect to our galaxy. You can see that in this 360 degree panoramic picture of the sky.
 |
| Click to see in full glory!!! via APOD. |
In addition to the familiar band of light from our milky way galaxy in this image, you might also notice the dimmer band of light running left-to-right. This is known as zodiacal light, and is caused by sunlight being reflected off tiny dust particles scattered like a pancake in the plane of our solar system. Zodiacal light can be used to literally see into the plane of our solar system. So here you can visibly see the plane of the milky way vs the plane of our solar system, and as you can see they are tilted with respect to one another by about 60 degrees.
You can see this even better in this image of the sky that was made by looking at infrared light (shifted here into the visible spectrum so we can see it, of course).
 |
| Infrared image of the sky. Click it! via APOD. |
The mega-bright line from left-to-right is the milky way galaxy. It looks more bright here than in the image above because infrared light can pass through the dust that normally blocks part of our view of the milky way. The more diffuse blue-ish light is the zodiacal light from our solar system. See, we are tilted by 60 degrees!
WOAH!!!!
*note: The tilt of the earth's axis of rotation with respect to the plane of the solar system and earth's orbit around the sun greatly complicate this whole issue. So, to avoid entering a quagmire of celestial-mechanical-insanity, I'm ignoring it entirely. Please forgive me. If you'd like to learn more, check this confusing stuff out. Also for a different explanation of all this, go here.