The rondeau redoublé is, as you can likely deduce from the spelling, a French form, which has been described as follows by the entertaining and educational Leonardo Malcovati:
this seriously minor, somewhat twisted and exclusively French metre, of which no more than a handful of examples (two of which, 'A Sylvie' and 'A Iris', of course, by Banville) exist, to show how twisted prosody can be, even in Europe.
Technically speaking a rondeau redoublé is made of six quatrains ended by a hemistich (of exactly the same type as the one in the rondeau form, and built on the first verse as well). The 24 verses, 4 of which are found twice (in the first stanza and as endings of stanzas 2-5) all belong to only two rhyme groups, one of which must be feminine and the other masculine; according to the usual conventions of this chapter, the tricky scheme of this form is:
ABAB BAB1 ABA2 BAB3 ABA4 BABAh
I'll bet I lost you at hemistich, right? I'll try to make it a bit simpler to follow than that technically correct (but presumes you speak poetic form language) definition.
Let's start with the name: rondeau redoublé, or "doubled round". The most famous of all rondeaux in the English language is In Flanders Fields by the Canadian poet, John McCrae, which you can read all about in this prior post of mine. The rondeau takes the start of the first line, usually three or four words (technically called a hemistich), and uses it as a refrain at the end of the following two stanzas - hence the repetition of "In Flanders fields" twice more in that poem. The rondeau does not require a particular number of lines per stanza, but usually comes in with three stanzas and a total of 13-15 lines.
The rondeau redoublé, like its simpler sibling, uses a form of refrain, and it also borrows from the start of the first line in order to end the poem. The rondeau redoublé, however, has rigid stanza and line requirements. It traditionally has six stanzas and a total of 24-1/2 lines to it. The first five stanzas all have four lines each; the last has four full lines plus the hemistich (the snippet from the start of the poem), thereby ending the poem precisely where it started (although hopefully having taken you somewhere else in the middle). The "refrain" in a rondeau redoublé is derived from the first four lines of the poem, each of which serves in turn as the last line of the next four stanzas. The final stanza goes its own way, but must end with that hemistich we talked about earlier.
Oh. And one more thing: the entire poem consists of only two end-rhymes. Traditionally, the first stanza uses ABAB rhyme, which means that stanzas two and four end with an A-rhyme, whereas stanzas three and five end with B. The last line of the stanza helps dictate the rhyme scheme to be used in that particular stanza - it may therefore rhyme BABA/ABAB or ABBA/BAAB, but whatever it does, it must end with its assigned line from the first stanza. The sixth stanza has to stick to the scheme, and must end using that hemistich.
Here are examples of three good rondeau redoublé in English for you, all of which are under copyright, and I've therefore sent you thither and yon to have a look at them.
1. Rondeau Redoublé by Sophie Hannah
2. Rondeau Redoublé (and Scarcely Worth the Trouble, At That) by Dorothy Parker
3. Rondeau Redoublé by Wendy Cope
You may, like me, have noticed that they are all entitled "Rondeau Redoublé" (Dorothy Parker's has a subtitle of sorts). That is not a requirement of the form, but I have my suspicions that the reasons for using it as a title lie among the following list:
1. It is a sort of warning. "Look," it says to the reader. "I know this is an unusual form, and I want you to know what it is."
2. It is a sort of apology. "I know there are only two rhymes and a bunch of stuff gets repeated. Sorry. I had to do it as a requirement of the form."
3. It is a sort of bragging. "Look," it says to other poets. "I have written one of these extraordinarily difficult poems."
4. It is a sort of exhaustion. "Hey, I wrote the damned thing, and asking me to stick a title on it on top of what I've just done is simply asking too much."