It seems that if the TARDIS is in normal flight, it has fixed up/down orientation.
However, if it is being acted on by some outside force (Angels, crash, impact with the Titanic), then the contents are affected by the movements of the TARDIS due to that force.
While I can't recall the TARDIS ever being turned completely upside down, or even on its side, it does not stay fixed vertically while flying, yet there is never any indication, on the inside, of the g-forces caused by its movements, or of any impingement by the external gravitational field.
This video shows what happens when 4 Weeping Angels tilt the Tardis back and forth.
The people inside are thrown about.
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Every room in the TARDIS apparently has it's own gravitational controls.
The clearest example of this is in
The Doctor's Wife when House has taken over the TARDIS (presumably looking for Vicodin?)
and Amy/Rory are running through the halls.
At one point House turns off the anti-grav on a room that is apparently tilted sideways, and Amy/Rory almost fall down a very long shaft.
However, they are able to climb into an adjacent room where gravity is normal.
It's less clear how River does her high-dive trick in Day of the Moon .
When the TARDIS matrialized on the side of the building, The Doctor and Delaware are still in the control room, which appears upright.
Amy and Rory have to go open all the doors to the pool, which I interpreted as meaning "open all the doors between the front door and the pool", but of course we never see River actually move through the TARDIS, so it's not clear what happens to her direction of movement when she goes through the doors.
My suspicion -- not really supported by canon but the most logical -- is something of a Portal effect.
Her vertical momentum was converted 100% into horizontal momentum the minute she passed through the door, and the TARDIS's fine gravity controls kept her aloft long enough to splash down in the pool.
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In season 19 "Castrovalva" episode 2/4, the tardis lands tilted and the control room was tilted, the companions and doctor all walked inclined to get out of Tardis.
BUT in 2005 series S07E10 "Journey to the center of the tardis" everything was level inside, after you walk in.
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In Journey to the centre of the Tardis it is made clear that the Tardis has its own gravity system.
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At the start of episode 1 of The Romans (1964), the TARDIS lurches violently and the floor tilts crazily, throwing the Doctor and his companions to the floor, as the TARDIS topples off the edge of a cliff and tumbles into a ditch.
In the 1968 serial The Web of Fear , episode 1 begins with the TARDIS tilted over on its side in the time vortex, lurching and off balance, because the doors have been opened in flight: the Doctor and his companions struggle to avoid being ejected through the open doors due to the sloping floor.
In The Curse of Peladon in 1972, the TARDIS lands half way up a mountain, and begins to topple over the edge of the cliff in a storm.
The Doctor and Jo have to struggle against the tilted floor to escape , in the moments before it falls into the chasm.
In Castrovalva (1982), as mentioned, the TARDIS lands on its side in episode 2 (at a 45 degree angle), because the newly regenerated Doctor is ill, leaving Tegan to attempt the landing.
In the novel: Tegan picked herself up from the floor, which was now leaning over at a crazy angle...
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Since the outside and the inside of the TARDIS are in two separate dimensions, the inside can be locked to the outside.
Meaning that if the outside rocks the inside rocks.
But it can be unlocked, so if the TARDIS was sideways the contents wouldn't necessarily be sideways.
Though most of the time it is probably locked.
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The Doctor can choose which room is connected to the Real World-interface (the main doors) and orient the said-room to match the situation.
An anti-grav mechanism is likely used.
But seeing as the TARDIS is old, and sometimes malfunctioning, the anti-grav can sometimes not work.
That's how River did it...
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The TARDIS has it's own gravitational field inside which would keep any of the contents (including lifeforms) fixed to the floor.
For more watch 'Journey To The Center Of The TARDIS' with matt smith and that explains the question in which you are asking better
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If the TARDIS is turned upside down or is laid horizontally, what happens to the contents - including people?  Does the TARDIS up / down orientation change? Or does it have an artificial gravity so that internal down is always down as experience by those on board?
What happens to the people inside when the TARDIS is tilted?
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<doctor-who><tardis><artificial-gravity>
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An anti-gravity mechanism is likely used. The TARDIS has it's own gravitational field inside which would keep any of the contents (including lifeforms) fixed to the floor. Also, The Doctor can choose which room is connected to the Real World-interface (the main doors) and orient the said-room to match the situation. When TARDIS is in normal flight, the orientation is fixed and an anti-gravity mechanism is used.
When TARDIS is in normal flight, the orientation is fixed. An anti-gravity mechanism is used.