Formulating a 2D world As is obvious from living in the world, time moves in a line progressing from the past to the future. There is one direction of causality that always moves forward. Tessimal thinks that two directions of causality, or planar time, is impossible without either having so much friction that nothing ever happens or events that happen randomly along a forward-only 1D path without warning. Also, most possible formulations seem to violate conservation of energy. And while this seems to be correct, such a hypothetical world doesn't prevent intelligent life from existing there necessarily.

This formulation, when tested with a simple one particle movement simulation, produces a world that balances creation and destruction of kinetic energy so that putting small amounts of energy will destroy it, while putting large amounts will create it in bulk. This world works with a series of discrete instants that depend on the ones above and to the left on the timeplane. The x and y fields denote the two past instants. 
# Controls the position of the particle given its two pasts
location = (past_particle_location.x + past_particle_location.y) / 2.0
# Controls the velocity of the particle given its two pasts
var velocity_added = (past_particle_velocity.x + past_particle_velocity.y)
velocity = (velocity_added / 2.0)
velocity /= 1.58
velocity *= abs(acceleration) + 1
	
velocity += acceleration
location += velocity
Here are the simulations run with an unreleased program by Tessimal. They simulate a particle travelling in a line, with forwards or backwards acceleration that can be adjusted. It's built to test out many different formulas by editing a small piece of code. For these examples, I'll treat the units as meters and the particle as a human trying to get somewhere on foot (presumably they've portaled into this world; the native inhabitants of this world would likely be more adapted to these weird physics). For the 2D world of my design, I'll use this physics system with the following edits:
Music Theory Music in planar time is much richer than in linear time.

Sine waves have two perpendicular frequencies. These would be processed by the ears in weird ways incomprehensible to us. One possibility is that they behave like two distinct unentangled aspects of pitch (which we can't comprehend). This would enable music to convey two moods at once, though perhaps in different ways per axis. I have a weirder hypothesis. When humans hear sound, they naturally do spectral analysis to separate pitches, but when tones have harmonics (which are different pitches), we typically hear it as one cohesive tone, BUT you can often still hear the component notes if you focus even slightly. It's not exactly the same thing, but I personally think (and choose to worldbuild) that the two pitches would be perceptible separately but also processed together as a more singular experience, and these two senses would be in parallel. (Kind of like how you hear each note in a melody, but the full thing is greater than the sum of its parts). A tone would probably be processed as its base note, heard both as two separate pitches and as one cohesive unit, plus a 2D grid of harmonics (allowing exponentially more timbres).

Also, it may seem like instruments would have difficulties producing notes with different horizontal and vertical frequencies, and that is kind of correct (a guitar string can only have one length). But, all that's needed to produce sound is vibration, and it seems likely that a string could vibrate like that under proper conditions (though length alone can't do that). If this isn't common enough, creatures wouldn't evolve specifically to detect it. However, the musical possibility is still there with digital technology, as it would likely confuse their senses enough to perceive something new, and digital speakers are basically just really precise vibration devices that could easily play asymmetric frequencies if adapted to this world.

Time signatures are really fun in 2D. In our music, a time signature has a certain number of beats (or smaller subdivisions) in each measure. These are typically differentiated by how the listener feels the groove, based on the downbeat and cycles/call-and-responses in the accompaniment. But it's a bit more complex than that; besides the number of top-level subdivisions there are also different ways of dividing it up further. For example, 12/8 is typically four triplet subdivisions, but it also could be six doublet subdivisions.

A planar time signature is based on a tiling of the plane, with a way of subdividing each tile. The only one we can remotely comprehend is the square tiling, where it can be separated into a vertical and horizontal time signature in 1D each. But they can be based on a triangle tiling with typical subdivision, a hexagonal grid with subdivision into other shapes, or irregular tilings. For some measure types, like octagons, they can't actually exist on their own, only in mixed meter. Although, some of these can in hyperbolic planar time, and in Euclidean planar time they can use a projection to emulate it (though this comes with tempo distortions.)
Color Since sine waves have two frequencies, light also has two frequencies. Perception of color is determined by detecting certain ranges of light frequency, and having multiple ranges in combinations. Our cones do red, green, and blue. Possibly, in order to gain the same versatility in a world with planar time, you would need nine cone types. Unfortunately though, unlike sound, it seems a bit unlikely that light would actually use these distorted frequencies unless it were perturbed