As shown in the picture above we are having a lot more writes then reads on our appserver. Cache is enabled and the Framework is running with PHP/Symofony and using HTTP Cache.
Is this I/O behavior normal? I doubt it, since with the cache the server should read more and write less.
Without a clear understanding of your usage pattern, it is simply not possible to answer your question with 100% accuracy.
However, please consider that:
in-memory caching generally lowers read iops for a cache-friendly workload
your write stream seems quite light anyway, so it hardly denotes a problem
If your cache is near-perfect and all your content is served from memory, you'll see hardly any read operations on your storage system.
Since write operations won't be cached to any large degree, your systems does need to write those to disk to make them persistent, those will then quite quickly dominate your IO.
That seems to be the case in your diagram. That does not necessarily need to be a problem though, it doens't look like write IO is a real bottleneck yet.
As to why you get write operations on a server:
atime) for each and every file access. That can add up. Use the noatime or relatime mount options to reduce that.