Gamera would have definitely improved any or all of the Mel Gibson historical epics (or any other Mel Gibson movie for that matter).
You’re right.
“Two kaiju enter! One kaiju leaves! Two kaiju enter! One kaiju leaves!”
I don’t get it either, it was a reasonable question.
And while an actual planet would be easily detectable through its gravitational effect on other planets, a small asteroid may not be so easy to detect. I’m not sure if we’ve actually mounted an astronomical telescope on an interplanetary probe and pointed it there. I think the only argument against it is that it’s not a stable orbit, and it would be incredibly unlikely that an asteroid just happened to get into that position and remained there for the past century or two.
They are. To really figure out what would happen if you had a counter-Earth, you’d need some sort of celestial mechanics computer simulation. If you have three objects interacting gravitationally, and one of them has a negligible mass compared to the other two, you have an example of the restricted three-body problem. That’s a lot easier to deal with than the full three-body problem. The full three-body problem is insoluble, so that’s why we need simulations.
MST3K: The theatrical version screened This Island Earth which also featured a Counter-Earth.
Pedantic correction: the general three-body problem (where all masses are non-negligible) is insoluble but there are specific cases of bodies with M1>>M2=M3 and M1=M2=M3 (and more generally M1=M2=…=Mn as in a Klemperer rosette) which do have explicit solutions. However, such solutions would require some very specific starting conditions for initial stability, and are readily perturbed and unstable for any reasonable periods of time, so we would not expect to find them in nature or in any system which is not artificially stabilized.
It is also worth noting that while the current orbit of the Earth has an eccentricity of only about 0.0167; while this is pretty close to being a circle, there would still be points at which Earth would lead or lad Htrae; probably not enough to see directly observe the other planet, but sufficient that other orbital perturbations (drag of the Moon, attraction of Jupiter) would eventually pull it out of synch. The orbits of Earth and Htrae would precess and over the long term (tens of millions of years) there is even the possibility that they might collide, or at least get close enough to disrupt one another’s orbit.
Stranger
Dr. Who’s “The Tenth Planet” episode also talked about a counter Earth (it was originally where the cybermen came from).
It’s a very old SF trope.
I’m sorry, Randy, it’s just that I view you as a pretty intelligent guy, able to quiz the basics of this out for yourself. 
So I wondered if the question weren’t more of a way to get a discussion going on the concept as you recalled it from movie/literature…
Metaluna, the other planet in This Island Earth is not a Counter-Earth, it’s extrasolar. CalMeacham can probably give more details.
[OT]My Luna cat has a new nickname now- Metaluna.[/OT]
It was Episode 305 - Stranded in Space that had a Counter-Earth.
Gamera is to movies as bacon is to food.
[Homer] Mmmm, bacon-flavored Gamera [/Homer]
To be perfectly precise, the Lagrange solutions require that M1 >> M2 >> M3. If any two of the masses are comparable to each other, you can get qualitatively similar solutions, but which differ numerically from the Lagrange points. I only learned this fact myself the hard way, when I had a problem on my comprehensive exams concerning a planet in a binary star system (M1 = M2 >> M3). I set up some trancendental equations, and was all set to show that they were solved by theta = 60 degrees, only to discover that lo, they weren’t.
Closer to the original topic, if you did try to put a mass on the opposite side of the Sun, it’d end up in a horseshoe orbit, like that of the asteroid 2002 AA[sub]29[/sub]. It moves a slightly closer (and thus slightly faster) orbit than Earth, until it gets close enough that it almost laps us, and then is moved outward into a larger (and thus slightly slower) orbit by perturbation from Earth’s gravity. 95 years later, when it’s almost fallen behind one lap, the Earth’s gravity pulls it back in to a smaller orbit, and the cycle repeats.
What are you, a realtor or something?
A few of my favorites improved:
Round Midnight
“You’re Gamera! You’re too good for this sh!t.”
The Conversation
“Gamera’d kill us if he had the chance.”
Hard Eight a.k.a. Sydney
“Gamera f&cked me, and now he’s gonna pay me what he owes me!”
All the President’s Men
“The whole thing was a Gamera operation.”
Yeah, I guess you’re right.
Now, when I read a fortune cookie fortune, I can see myself adding, “with Gamera” at the end. 
You will find good fortune … with Gamera.
I believe the last two season of Lexx involved a counter-Earth.