Yes, both those pushes are valid.
When you push a creature, each square you move it must place it farther away from you. [RC 211)
Now, 4e measurements use taxicab geometry, or Chebyshev distance, so the effect is... often nonintuitive. I'm pretty sure 4e floors are constantly shifting hyperplanes. My players have developed the mantra, "Circles are squares, squares are circles, and never draw a triangle on the grid!"
How to tell if your push is valid:
To tell if a creature is moving closer, further, or the same distance, count the number of squares the creature must enter to be adjacent to you if he starts in the first square, or the second. The more squares he must traverse to reach you, the further he is from you.
You also need to have line of effect to every square you push him into. He can't enter blocking terrain, and ignores difficult terrain.
The flavor text describing a power's visual appearance has no effect on the power's mechanical application.
Weird.
This actually means that the squares you can legally push a creature into differ, depending on if the creature is head-on with your square or adjacent by corners, because any square adjacent to both you and him is invalid for a push when he is also adjacent to you.
However, so long as you fulfill the above prerequisites, you're free to bounce the guy around zig-zag-style; nothing says the movement needs to describe a straight line (and given 4e geometry, I'm not sure what a straight line would look like anyway).
For completion's sake, pulls and slides:
For pulls, each square must be closer to you than the last, and slides don't care what your position is related to the target.
Both, one, or neither
The case for using both
Using both may seem counter-intuitive and maybe even against the rules. I will address both these issues.
What's the point of moving a creature ten feet toward you and then ten feet away from you? Doesn't that just cancel out any movement? Yes and no.
There are some situations when moving a creature ten feet toward you and then ten feet away from you, or the other way around, could be very useful. For example:
A goblin is standing on a trap door that is triggered when you step off the floor plate. You hit the goblin with Eldritch blast, push the goblin ten feet away, the trap door opens, and you pull the goblin ten feet towards you over the hole where the trap door just opened. Bye bye goblin.
An ogre is ten feet in front of you. Your fighter companion is ten feet behind you ready to attack. You hit the ogre with eldritch blast and pull the ogre ten feet toward you and then push the ogre ten feet, the other way, so the ogre is in striking distance of your fighter companion who may proceed to smite the ghastly beast.
There is some controversy in using both actions as both Repelling Blast and Grasp of Hadar state they can be used When you hit with the eldritch blast. Does this mean at the precise moment when you hit? So you can only use one or the other at that precise moment, or does it allow for both to be used consecutively right after you hit? I would say that is up to the GM, unless there is a specific ruling I'm not aware of. As GM, I would rule you can use them consecutively immediately after you hit with Eldritch Blast.
V2Blast points out that this is in line with the optional rule on Simultaneous Effects in Xanathar's Guide to Everything (XGE 77):
If two or more things happen at the same time on a character or monster's turn, the person at the game table - whether player or DM - who controls that creature decides the order in which those things happen.
The case for using either one, or neither
The other options: to choose push or to move towards you, or to use neither, are supported by the word can in the description of both features. You can push and you can move. This means you have the option to use that feature or not. There is nothing that states they must be used.
Repelling Blast (PHB 111):
When you hit a creature with eldritch blast, you can push the creature up to 10 feet away from you in a straight line.
Grasp of Hadar (XGE 57):
Once on each of your turns when you hit a creature with your eldritch blast, you can move that creature in a straight line 10 feet closer to you.
Best Answer
One mistake: Lance of Lethargy can only be applied once per turn.
You write:
However, Lance of Lethargy states:
So you can only reduce the target's speed by 10 feet once in a turn.
Everything else works as described. Notably, spike growth does not require willing movement:
It only requires that a creature move within the area, the method and intent of movement are not relevant.
Now, you ask about the balance of this combo - it is balanced by the fact that it takes two turns to set up and your enemies are (usually) not stationary. It is a strong combo if it works, but since it will (almost always) require eldritch blast to be cast on your turn after you cast spike growth, there will often be time for the target to do some maneuvering to diminish the effectiveness of the combo. I say almost always because there are some dirty fighter multiclassers out there using Action Surge to cast two action spells in a turn.