Is the following possible without storing pointers?

<Not that I have something against pointers, but they'd make the class awkward to use>

The problem is about a matrix class.
I'm trying to make a trade-off...which I think is justified.

The idea is to store both the matrix and its transpose as member vector<vector<T>> `s. (Not exactly T...a pointer..reference..or something else)
That way... transpose would just be a std::swap and grabbing or interchanging rows and/or columns would just be accessing or swapping a vector<>.

Forgetting about templates for now, consider I have an integer matrix;
How exactly should I design it, such that, when an element from either of the representations get modified, the other one gets changed to, automatically ?
(This felt like a simple thing to do at first.....and here I am...after some blunders!)

Well since you can only efficiently swap either rows or columns, you need to make a choice.

Or you could do something like this.

Code:

vector<vector<T>> matrix;
vector<vector<pair<int,int> > > transpose;

If matrix[row][col] is an element,
it's transposition is at matrix[transpose[row][col].first][transpose[row][col].second]

By suitably initialising all the pairs, you can achieve any arbitrary matrix transformation.

I feel that you're trying to go about it the wrong way. You don't have to store it both ways around, and I don't see why you'd want to store vectors internally. All you'd need to do is to make your interface capable of allowing the accessing of the members to be in row or column major order.
I.e. try out the following code which for this example just stores ints and the matrix is 4x4 in size.

Code:

class constProxyAccessor {
	const int *base, step;
public:
	constProxyAccessor(const int *base, int step)
		: base(base), step(step) {}

	int operator[](int x) const {
		return base[x * step];
	}
};

class proxyAccessor {
	int *base, step;
public:
	proxyAccessor(int *base, int step)
		: base(base), step(step) {}

	int& operator[](int x) {
		return base[x * step];
	}
};

class transposableMatrix {
	int m[16];
	bool inverted;
public:
	transposableMatrix() : inverted(false) {}
	void invert() {
		inverted = !inverted;
	}
	constProxyAccessor operator[](int x) const {
		return inverted ? constProxyAccessor(&m[x*4], 1) : constProxyAccessor(&m[x], 4);
	}
	proxyAccessor operator[](int x) {
		return inverted ? proxyAccessor(&m[x*4], 1) : proxyAccessor(&m[x], 4);
	}
};

int main()
{
	transposableMatrix mtx;
	mtx[3][1] = 123;
	mtx.invert();
	int foo = mtx[1][3]; // foo is now 123!
	mtx[2][0] = 456;
	mtx.invert();
	foo = mtx[0][2]; // foo is now 456!
}