dynamic_cast back to same object type fails with multiple inheritance and intermediate variable

Suppose a hierarchy with two unrelated polymorphic classes PCH and GME, a subclass PCH_GME : public GME, public PCH, and an object gme_pch of type PCH_GME*.

Why does the following sequence of casts of gme_pch "break" the cast back to the object's original type GME_PCH*:

GME_PCH *gme_pch = new GME_PCH();    
GME *gme = gme_pch;
PCH *pch = (PCH*)gme;
GME_PCH *same_as_gme_pch = dynamic_cast<GME_PCH*>(pch);
// same_as_gme_pch is NULL

whereas the following code does not break the casts:

GME_PCH *gme_pch = new GME_PCH();    
PCH *pch = gme_pch;    
GME_PCH *same_as_gme_pch = dynamic_cast<GME_PCH*>(pch);
// address of same_as_gme_pch == gme_pch

Question: Isn't each pointer always pointing to the same object, and shouldn't then the result of the final cast back to the original type always be the same?

---

Just for making it easier to try things out, see the following code demonstrating variants of the above mentioned cast sequences; some working, some not:

class PCH {  // PrecachingHint
public:
    virtual std::string getHint() const = 0;
};

class GME {  // GenericModelElement
public:
    virtual std::string getKey() const = 0;
};

class GME_PCH : public GME, public PCH {
public:
    virtual std::string getHint() const { return "some hint"; }
    virtual std::string getKey() const { return "some key"; }
};

void castThatWorks() {

    GME_PCH *gme_pch = new GME_PCH();

    PCH *pch = gme_pch;

    GME_PCH *same_as_gme_pch = dynamic_cast<GME_PCH*>(pch);

    std::cout << ((same_as_gme_pch == nullptr) ? "cast did not work." : "cast worked")<< std::endl;
}

void castThatWorks2() {

    GME_PCH *gme_pch = new GME_PCH();

    GME *gme = gme_pch;
    PCH *pch = dynamic_cast<PCH*>(gme);

    GME_PCH *same_as_gme_pch = dynamic_cast<GME_PCH*>(pch);

    std::cout << ((same_as_gme_pch == nullptr) ? "cast did not work." : "cast worked")<< std::endl;
}

void castThatDoesntWork() {

    GME_PCH *gme_pch = new GME_PCH();

    GME *gme = gme_pch;  // note: void* gme = gme_pch breaks the subsequent dynamic cast, too.
    PCH *pch = (PCH*)gme;

    GME_PCH *same_as_gme_pch = dynamic_cast<GME_PCH*>(pch);

    std::cout << ((same_as_gme_pch == nullptr) ? "cast did not work." : "cast worked")<< std::endl;
}

void castThatDoesntWork2() {

    GME_PCH *gme_pch = new GME_PCH();

    GME *gme = gme_pch;
    PCH *pch =  reinterpret_cast<PCH*>(gme);

    GME_PCH *same_as_gme_pch = dynamic_cast<GME_PCH*>(pch);

    std::cout << ((same_as_gme_pch == nullptr) ? "cast did not work." : "cast worked")<< std::endl;
}

void castThatDoesntWork3() {

    GME_PCH *gme_pch = new GME_PCH();

    GME *gme = gme_pch;
    PCH *pch = static_cast<PCH*>(static_cast<void*>(gme));

    GME_PCH *same_as_gme_pch = dynamic_cast<GME_PCH*>(pch);

    std::cout << ((same_as_gme_pch == nullptr) ? "cast did not work." : "cast worked")<< std::endl;
}


int main() {

    castThatWorks();
    castThatWorks2();
    castThatDoesntWork();
    castThatDoesntWork2();
    castThatDoesntWork3();   
}