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: 
Probation periods tend to allow *either* party to give shorter notice (not just the employer).

And you can certainly either ask or insist that they change it so it goes both ways.

However:

* This might be something the company is unwilling to change, especially if they're covering your relocation expenses.
* It might indeed make you seem a bit undecided or unreliable, or perhaps like you're trying to take advantage of them (e.g. say they're providing sponsorship and said sponsorship would be easily transferable or would last sufficiently long after you've stopped working there).

As such, asking (and definitely also insisting) that it be changed can certainly put the offer at risk.

Although I wouldn't expect just asking to be particularly risky - if they're unwilling to change it, I'd expect the most common response to just be for them to tell you that and not withdraw the offer (if the employer is reasonable, that is).

As an alternative to proposing that it be the same for both sides, you can also consider proposing a notice period somewhere between 1 week and 6 months.

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I'm assuming they're not doing anything illegal by proposing this one-sided probation period - as others pointed out, that might not be the case.

The 6 month notice period also seems excessive long to me, but I wouldn't be able to comment on whether it's the norm in your industry.
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.