Does ATC direct flights to fly eastbound at odd levels and westbound at even ones?
In addition to ymb1's excellent answer, here are a few thoughts from the controller perspective.

Various rules and regulations for cruising level allocations exist. For flightplanning purposes, the semicircular rules are very notable. As pointed out by ymb1, the exact rules vary from place to place. In some regions, flights flying on a westbound track shall plan to fly at even levels while flights flying eastbound shall plan to fly at odd levels. Note that this is in relation to the magnetic track of the aircraft, except in areas near the poles. In some countries it is the other way around, in others, the rule is north/south instead of east/west.



Example of a general east-odd/west-even rule (excerpt from AIP Denmark)

You can also have specific ATS routes, such as airways, that require you to flightplan certain levels that deviate from the general rule.



Example of ATS route which enforces even levels for eastbound flights (excerpt from AIP Germany)

It is important to note that rules for level allocations are mainly used for planning purposes. When filing a flightplan, the planned levels must obey any such rules - and you should always stick to your filed flightplan when in uncontrolled airspace.

However, when being provided with air traffic control service, the controller may deviate from the general rules if so desired (unless the local operating procedures do not allow it). If I have an upcoming conflict that can be solved by climbing a westbound flight to FL370, then I will be more than happy to do so - especially because establishing vertical separation is often the easiest way of solving a conflict. As long as the flight is within my sector, then I make the rules, so to say.



Example of ATC radar screen with eastbound traffic (SAS528) flying at an even level (FL340) for separation

When transferring a flight to the next sector (or another unit), the controller is responsible for making sure the flight is at an appropriate level in accordance with the local letters of agreement. Principally, this means that we have to go back to the published rules (east odd, west even or specific airway rules). In practise, though, if it is desired to transfer a flight on a non-standard level, all it takes is a phone call to the next sector to check if they can accept it. If the next sector has no conflicting traffic, then there is nothing wrong with keeping the flight at a non-standard level.

So to sum it up, when planning a flight, you have to obey the published level allocation rules, which vary from place to place. A large part of the world uses the east-odd, west-even rule. But once under ATC control, there is no need to worry if you are assigned a non-standard cruising level - if that happens, it is because it is allowed and has been coordinated.