Pseudofly
Pseudofly (or pseudo) is an advanced hammerfly technique similar to Speedfly in which the two tees never come into contact, but enables much more dynamic movement through the air when done correctly. With enough practice, a tee (or tees) can pseudofly up, down, and sideways, at will. Pseudofly is generally easier with a dummy, but can also be done with another player as well, although requires great coordination between the two.
Hook timing
The important difference between pseudofly and other Hammerfly is the hook timing. Rather than hooking every two hammer hits, the driver must hook every one 
Hammer hit. The main focus of the driver is to hook just long enough to get into hammer distance. If the driver hooks too long, the hammer will still be in cooldown when two tees collide and turn into a normal hammerfly.
The driver must constantly be paying attention to the distance between the two tees to judge how long to hook for each hammer hit. Once you get comfortable with the hook timing, you will be able to move in all directions at any speed you wish - or even stay in one spot. A general rule of thumb is to hook more when going faster or going sideways.
Flying down
When pseudoflying down, the driver must hook for shorter time lengths as well as more sparingly. Flying down is basically like falling - except occasionally, the driver hooks and the hammer tee hammers.
When leveling out again, simply increase the rate of hooking and hammering.
Transitioning into speedfly
To transition into speedfly, the driver must be especially careful about the hook timing. When you feel yourself starting to go up, the driver must hook for a little bit longer as speed increases. Once you go fast enough, the driver must hook for every two hammer hits instead of one, like a normal speedfly.
An alternative method to transition to speedfly is to have both tees hammer each other at the same time, and the driver to quickly move left or right, and then level out again. This is much faster method, although the same principles as above still apply.
Hammer timing
Although the majority of pseudofly is dictated by the driver, a good hammerer can complement the driver. If the driver is very competent, the hammerer can spam hammer, and the driver can still take full control, but that is too not realistic for every driver.
Basically, hammerer should time the hammer before the two tees collide and preferably be consistent with hammer timing. Similarly to hook timing, hammer should be more frequent when the momentum is large, or an acceleration is desired, and vice versa. There are a few ways in judging the hammer timing:
By distance
Hammer's sprite can be a good reference point. The tip of the hammer is roughly the maximum range and hammer at half the distance is idealistic with room for adjustment. The hammerer can adjust the distance accordingly. For example, the hammerer can hammer when two tees come very close when they are descending, so a minimal upward momentum is gained.
By rhythm
If the hammerer is familiar with pseudofly, the tee can hammer with a certain frequency, and hammer faster or slower according to movement changes. This is less adaptive to sudden changes and requires higher driving skills, but less tiring than timing every hammer by distance.
Path prediction
The hammerer should judge where and how the driver wants to go, and hammer accordingly. This requires a high level of coordination and a high map reading skill. This works well when there is a clear path of flying, but not for random flight over a plane.
A competent hammerer should also be able to adjust their hammer timing especially for tight configurations. When going sideways, the hammerer can press movement key to increase speed and height.
Use cases
A common use case for pseudofly is to fly through tight horizontal corridors that would be unpractical, or impossible, for any other kind of fly.
Pseudofly is also a common technique used in speedrunning. It can be used to preserve a large speed, usually comes from a big swing.
Starting a pseudofly
When pseudoflying with a dummy, it's important to control the rate at which the dummy hammers. Therefore, auto-hammer is not enough. You can bind a key to have your dummy hammer at will using the following command in the console (F1):
bind x "+toggle cl_dummy_hammer 1 0"
Once you have that set, you can start pseudoflying.
It is possible to start pseudofly in the air, but it is easiest on flat ground. Start by standing next to the tee, then have the dummy hammer you at a fast rate, hook him at the same time, and start moving in the opposite direction in which you are positioned from the dummy.
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Starting pseudofly from the ground with dummy.
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Starting pseudofly midair with dummy.
When pseudoflying with another player, it's important to communicate with them when it should begin. As the driver, you can do so by hammering, switching to the pistol, and shooting at a fast rate.