Question: I have received a very good job offer at a company in finance after a long application process.

From the first day onward, I would be required to give notice of termination 6 months prior to the termination date, and I am entitled to receive the same. This seems common in this field.

However, there is a 3-months probation period, during which my employer would be allowed to terminate the contract with one week prior notice, but I don't get the same right. 
I was assured that it never happened that the company actually made use of the probation period rule, and it also never happened that someone wanted to quit in the very beginning (which actually does not surprise me, they seem great). 

On the one hand, I feel it would not be so much to ask to include the week termination period for me as well, since there seems almost no risk that I would make use of it. I am very certain I will like it there. On the other hand, I would like to minimize risks.

Is it strange to ask for this? Would it sound as if I am not sure or I am afraid I might get cold feet?

**Addendum:** I would relocate from the US to Europe for this, this would show a certain level of commitment from my side as well.

Answer: 
The probation period in several European countries works both ways. 

*Usually* contracts also cannot subtract rights that are written in the law, and such clauses can be indeed deemed illegal in several countries. 

I would never accept a one-sided probation period nowadays. On the other hand, I was an expat a few years, and I know first hand that it can be customary to apply one-sided probation periods for people going abroad for covering their investment. However I was working in Africa, and I am confident quite a few European countries enforce a dual sided probation period. I would let it go and accept the probation period because you need to focus on other issue.

A six month notice period is pretty abusive, and again almost all the same comments apply, excepting that is not that common. In practice, if enforced, it can deter you from changing positions. Most employers arent willing to wait six months for a new prospective hire. Try to negotiate it down to at least 3 months if it is not common to your industry.

You are not obliged to accept all the terms on a proposed contract and you can and are expected to negotiate it; if in doubt, talk with a lawyer specialised in European work law.

Be aware how the six-month notice clause is worded also. Depending on the wording, it might prevent you for getting another position for a significant period of time in the remote possibility they use the probation period. Or it might hide a non-compete clause in some other legalese.

Ask for a copy of the contract(s) *before signing them* and show them to a lawyer instead of asking random people in the Internet.

I also agree with the comments bellow the biggest no-no on this contract is the lengthy notice period; do not sign anything you might regret on your future.

If hired in the US by a firm with an American subsidiary:

* You probably will have an US *and* you must have an European contract;
* ask for both legs of the contract to show a lawyer, both the US and the European one;
* Do not believe the white lie the European contract "will be just a formality";
* Beware of clauses in the US contract deemed to get around rights given by European labour law.

I would consult a lawyer if I were on your shoes; I did not before signing my expat contract, being younger and more naive, and had it gone wrong I would be in big troubles (yeah, that kind of clauses "it never happened to us"...it sure did happen before, people gone sick, people that did not adjust well to the new country, people gone berserk with the new environment getting into drugs and drinking as I found out later through the grapevine ). 

Finally, a prospective employer talking you down about clauses of a contract is an huge red flag that should not be ignored. Remember, HR are not your friend.
In Germany a 6 month probation period is the maximum allowed in most circumstances, but that period is also quite common for most jobs. However it is not at all a requirement, and it is possible to negotiate a shorter or even no probation period at all. Waiving a probation period is really uncommon (and not sure why you would want that anyway), but there is nothing legally against it.

There is no circumstance where an employer can legally demand a shorter notice period than the employee gets. 

During the probation period both employer and employee may terminate with a two weeks notice - a shorter notice period is forbidden by law. After the probation period the notice period for the employee is a minimum of 4 weeks to the 15th or end of the month, same for the employer. A 3 month notice period is not at all unusual for skilled employees, and prospective new employers don't bat an eye if they have to wait 3 months before you can join. etc etc...any point in continuing?

*Europe is not a country* Please provide a country tag as customs and laws vary quite a bit from place to place.