Unfortunately, in my opinion, people wrongly assume there's a one-to-one correspondence between the first 3 and last 3 blessings. I think a parallel exists between "shevach" and "hodaah" - but only if "hodaah" is taken to mean the single blessing of Modim. Modim contains 3 distinct sections with separate themes, each of which is linked to one of the first 3 "shevach" blessings.
Modim is significantly longer than any other blessing in the Shemoneh Esreh. The entire Shemoneh Esreh contains 642 words (in the nusach I found online), while Modim contains 94. Do the math and you will find that the other 18 blessings are on average 30.4 words long, so Modim is 3.09 times as long as the average. Thus supporting, perhaps, the idea that Modim is meant to parallel 3 other blessings.
In order to identify the possible sections of Modim, let's start with the obvious. According to the punctuation I will use below (scroll down if you want to see it), Modim contains 6 sentences before the concluding "baruch". Logically enough, I decided to divide these 6 sentences into sections of 2 sentences each.
One striking consequence of this division is the location of the key word "modim/lehodot", which appears 3 times in the blessing, not counting once in the conclusion. The first section begins with "modim", the second section begins with "nodeh", and the third section contains the word "yoducha". As its name indicates, Modim is about "hodaah", and each section mentions "hodaah" once, in two out of three cases right at the beginning. This is strong evidence that the division is correct.
Looking more closely at each section, we see that each has a very distinct theme from the others. Not only that, but these themes closely match the themes of the first 3 blessings of Shemoneh Esreh. This is best shown on a section-by-section basis.
Modim begins by thanking God for being God. This unusual beginning happens to match the very beginning of the Shemoneh Esreh, virtually word-for-word. Instead of "Blessed are You, Hashem our God and God of our ancestors", we have "We thank You, Hashem for being our God and God of our ancestors". Later on, the phrase "guard who saves us" is virtually the same as the phrase "savior and guard" from that first blessing (Avot). We therefore see that the opening third of Modim consists mostly of verbatim quotes from Avot.
There are also deep halachic parallels between Avot and the first section of Modim: both involve bowing down, and both Avot and (according to some opinions) Modim invalidate Shemoneh Esreh if said without concentration [kavanah]. Neither bowing nor concentration is necessary for any other blessing. (The other blessings must of course be said with kavanah, but lack of kavanah does not invalidate them.)
What is the point of the linguistic and halachic similarities? The purpose of Avot is to invoke and initiate our relationship with God, which is a vehicle for the prayer which we are beginning. In Modim, as we approach the end of the Shemoneh Esreh, the point is apparently to thank God for the relationship which allowed Shemoneh Esreh to be said.
Both Avot and the beginning of Modim are therefore "prayers about prayer". If you don't have kavanah in (for example) the blessing regarding rain, who knows, you may not merit to get rain. But if you lack kavanah in Avot or Modim, you never properly accepted upon yourself the basis for prayer, so your entire Shemoneh Esreh becomes invalid.
מודים אנחנו לך שאתה הוא ה' אלהינו ואלהי אבותינו. צור חיינו, מגן ישענו אתה הוא לדור ודור.
ברוך אתה ה' אלהינו ואלהי אבותינו, ... מלך עוזר ומושיע ומגן. ברוך אתה ה' מגן אברהם.
This section differs from the first section, in that it does not seem to talk about our relationship with God, but about various individual deeds which God does for us. In particular, it recognizes God's sustaining us, providing life and performing miracles for us, on both a short-term and long-term basis.
Unlike with Avot, there is no exact verbal duplication of Gevurot in Modim. But the theme of Gevurot and this part of Modim are very similar. Both consist of lists of the various kindnesses which God performs for us. The lists are not identical (though both focus particularly on the continual sustaining of our lives). But the main point is not the choice of examples, but the fact that such examples exist. Whatever the specifics - and on different occasions, we approach God regarding different specifics - we acknowledge that God is the correct address for such inquiries.
נודה לך ונספר תהלתך, על חיינו המסורים בידיך, ועל נשמותינו הפקודות לך, ועל נסיך שבכל יום עמנו, ועל נפלאותיך וטובותיך שבכל עת, ערב ובוקר וצהרים. הטוב כי לא כלו רחמיך, והמרחם כי לא תמו חסדיך, כי מעולם קוינו לך.
אתה גבור לעולם ה', מחיה מתים אתה רב להושיע. משיב הרוח ומוריד הגשם. מכלכל חיים בחסד, מחיה מתים ברחמים רבים, סומך נופלים ורופא חולים ומתיר אסורים, ומקיים אמונתו לישני עפר. מי כמוך בעל גבורות ומי דומה לך, מלך ממית ומחיה ומצמיח ישועה. ונאמן אתה להחיות מתים. ברוך אתה ה' מחיה המתים.
It's also worth noting the dual introduction to this part of Modim: "We must thank you, and recount your praise, for..." There are two distinct elements here: recognition and thanks. We are listing examples, and also expressing our feelings about each example.
One of the things we thank God for is "your wonders and good deeds which occur at all times - evening, morning, and afternoon." I think it's no coincidence that evening/morning/afternoon are the 3 times at which we pray. We are saying that whenever we pray, God listens to and responds to our prayer. "Vehaya terem yikreu va'ani aaneh".
In this section, in contrast to the previous ones, the focus is not God or His deeds, but rather on our response to those deeds. We hope to praise God and recognize what he does for us. We hope to do this both constantly and forever, and expect that eventually all of humanity will join us. The idiom is that we praise not God, but rather His "name" or reputation. This emphasizes even more that we are focusing not on God, but on ourselves, on our own comprehension of and relation to God.
Virtually the same description applies to the third blessing of Shemoneh Esreh (Kedushat Hashem). Here too the focus is on our response: the constant praise and recognition of God, forever, involving all humanity.
ועל כולם יתברך ויתרומם שמך מלכינו תמיד לעולם ועד.
וכל החיים יודוך סלה, ויהללו ויברכו את שמך הגדול באמת לעולם כי טוב, האל ישועתינו ועזרתינו סלה, האל הטוב.
[ברוך אתה ה', הטוב שמך ולך נאה להודות.]
לדור ודור נגיד גדלך ולנצח נצחים קדושתך נקדיש...
אתה קדוש ושמך קדוש, וקדושים בכל יום יהללוך סלה, כי אל מלך גדול וקדוש אתה.
ויראוך כל המעשים, וישתחוו לפניך כל הברואים... ושמך נורא על כל מה שבראת. (ימים נוראים)
It appears that there are 3 distinct sections to Modim, which are very similar to the 3 "shevach" blessings at the beginning of Shemoneh Esreh.
Interestingly, Shemoneh Esreh as a whole has 3 sections as well. The first section is "shevach", the last "hodaah", and the middle consists of all the requests in the other blessings. Each of these sections corresponds to one of the 3 "shevach" blessings, as well as one of the 3 sections of Modim. That is to say, the structure of the first 3 blessings matches the structure of Modim, and both structures match the overall structure of Shemoneh Esreh.
- The "shevach" unit serves to introduce our prayer. Avot itself introduces our authority to pray, and the first third of Modim is a thank-you for being able to pray.
- The middle of Shemoneh Esreh is a series of specific requests. Gevurot lists the specific things God does for us, and the second third of Modim mentions those same things and thanks God for them.
- The "hodaah" unit concludes the prayer by acknowledging and thanking God for what we receive from Him. Kedushat Hashem and the last third of Modim describe our thanking God, both now and in the future.