This is a tough one. I am not sure I know. But it has to do with the impersonal usage of Turkish passive which is a special feature of it. Of course I would miss it completely is there wasn´t a similar thing in Finnish. That´s why I understand completely when scalpel says

         yıka|n|ıl|dı = ´wash´ + ´himself´ + ´is done´ + ´sometimes in the past´.

(Well, this is not what he said but this is how I understood it.)

In Finnish it would be

         pese|ydy|t|ti|in = ´wash´ + ´self´ + ´is done´ + ´sometimes in the past´ + poss sg 3rd,

which means some people (probably more than one) washed themselves in the past. This is our passive. It can be formed from intransive verbs as well as transitive. It is used for reflexives as well. When we use impersonal passive we always mean that some people did the action, we just don´t know who.

Geoffrey Lewis writes: "The most remarkable feature of the Turkish passive is its impersonal use: niçin yalan söyle-n-ir? ´why are lies told?, i.e. ´why do people tell lies?´ In this example the passive verb appears to have a subject, but impersonal passives are also regularly formed from intransitive verbs and then have no conceivable grammatical subject..."

I also like the Wikipedia article about impersonal passive. There are some Turkish and German examples there.

Taking the risk of talking to myself I still want to stress that what I just wrote was only an answer to si++´s question. Yes, in world languages reflexive and passive markings can live just fine in the same verb when we talk about impersonal passive. And passive voice does have impersonal use in Turkish. That´s all.

What comes to the original question about what these odd verb forms really consist of naturally I won´t even try to answer it. It is for the natives to take or leave.

Well, I have given a thought to it and have some explaination now.

söylendi = he told something(s) to himself (reflexive but we have another object here)

so

söylenildi = something(s) was/were told by him to himself.

yıkandı = he washed himself (reflexive)

now let´s try with some adverbs in it.

nehirde yıkandı = he washed himself in the river

now we don´t say something like this:

nehir yıkanıldı (instead we say "nehirde yıkanıldı")

but impersonally

(içinde) yıkanılan nehir = the river in which they wash themselves

(or liretally the river in which they are washed by themselves)

According to the definition ("decreases the valency of an intransitive verb to zero")

         (Nehirde) yıkanıldı

is an impersonal passive.

The red text is not clear to me. Where is that definition from?

Nehirde yıkanılır may be impersonal and should mean:

one washes oneself in the river.

But when I say "nehirde yıkanıldı" I think I point out to a specific event performed by known (specific) people and it should mean:

they were washed by themselves in the river.

Or

şu anda nehirde yakanılıyor = they are washed by themselves in the river at this moment.

What exactly do you mean by impersonal passive?

Edit:

Never mind I have located the wikipedia page:

http://en.wikipedia.org/wiki/Impersonal_passive_voice

You can find the definition for instance in the Wikipedia article

         Impersonal Passive.

The valency of a transitive verb is usually two: it needs a subject and an object. The valency of an intransitive verb is usually one: It doesn´t need but a subject. The definition talks about this latter type. Impersonal passive changes an intransitive verb (whose valency is normally one) so that it doesn´t need either a subject or an object to be a grammatically acceptable sentence. Just like yıkanıldı in your example.

The meaning of an impersonal passive is different from the classical Latin passive. When impersonal is used it always means that some people (the speaker doesn´t know or care who) are performing the action. Always some people. Just like you described the meaning of your sentence.

I don´t claim Turkish passive is completely an impersonal but it has usage which can be defined impersonal.

I remember my professor always started his passive lectures with the same words: "Remember, in Finnish (sorry, again) we don´t actually have a passive, we have an impersonal." At this point I was fighting to keep my eyes open. Only after beginning to study Turkish I actually understood what the old man was saying.

Everytime I learn something I realise how little I know!

Unergative verb, unaccusative verb etc.etc.

However the Turkish example from the wikipedia page, which is said to be invalid sounds OK to me.

The verb ölmek "to die", however, is unaccusative and may not be passivized:

*Burada öl-ün-ür. (I think this is OK. It should mean "this is the place where you can die")

here die-PASS-PRESENT

"Here it is died."

Yep, it´s beautiful. New worlds open.

But yıkanmak is not an intransitive verb but reflexive. I would think it as a (special kind of) transitive where the object = the subject. The English translation helps to reveal the object:

yıkandı = he washed himself

A reflexive has a logical object, of course, but still the syntactic valency is like that of any intransitive verb. One. In agglutinative languages the reflexive marking (sort of) makes transitive verbs intransitive from the syntactic point of view.

It is a real pleasure to discuss these things with someone like you who has a great skill in language/grammar..  

yıkanıldı has exactly the same meaning as peseydyttiin does in Finnish: ´some people (probably more than one) washed themselves in the past.´

To the contrary of what some people may think, reflexive and passive are different things and a reflexive verb can have its passive form when necessary.

Remember the example I gave in my previous post:

a) Böyle de giyinmez ki (reflexive)

b) Böyle de giyinilmez ki (reflexive+passive)

This example explains well that reflexive+passive in Turkish  has its field of use. 

We also have verbs containing reflexive+passive combination:

aranıl(mak), yüklenil(mek),etc. 

..and an adjective: 

kaçınılmaz

Maybe there are a few more but these are what I remember at the moment.