My work is based upon that of Bernard Jancewicz, e.g.:

The extended Grassmann algebra of R3.
Clifford (geometric) algebras (Banff, AB, 1995), 389--421,
Birkhauser Boston, Boston, MA, 1996

Here is a brief description of the relationships
between Jancewicz's work and my algebra. I will
exhibit a table (in two parts). Here e_x, e_y, and
e_z constitute an orthonormal basis of vectors for
three-space with a Euclidean metric to be definite.
Also where Jancewicz uses the term odd (previously,
pseudo-) I use twisted. Similarly, where he uses the
term even (previously, no prefix) I use straight.
Analogous to the notation of differential geometry I
symbolize the unit straight positively-oriented trivector 
as an upper case Omega = e_xyz = e_x /\ e_y/\ e_z. 
This table introduces two new representations
for directed quantities.


     Directed       |  Jancewicz  |  Differential
     Quantitiy      |  (modified) |    Geometry 
------------------------------------------------------
Straight Scalar     |      1      |   1
Twisted Scalar      |      r      |  (1, Omega)
Straight Vector     |      e_x    |   e_x
Twisted Vector      |    r e_x    |  (e_x, Omega)
Straight Bivector   |      e_xy   |   e_xy
Twisted Bivector    |    r e_xy   |  (e_xy, Omega)
Straight Trivector  |      e_xyz  |   e_xyz
Twisted Trivector   |    r e_xyz  |  (e_xyz, Omega)


     Directed       |     Extremum    |  Correlated 
     Quantitiy      |    Grade Form   |  Grade Form 
------------------------------------------------------
Straight Scalar     |  [1, 1]         | [1, 1]
Twisted Scalar      |  [Omega, 1]     | [Omega, 1]
Straight Vector     |  [1, e_x]       | [e_x, e_x]
Twisted Vector      |  [Omega, e_x]   | [e_yz, e_x]
Straight Bivector   |  [1, e_xy]      | [e_xy, e_xy]
Twisted Bivector    |  [Omega, e_xy]  | [e_z, e_xy]
Straight Trivector  |  [1, Omega]     | [Omega, Omaga]
Twisted Trivector   |  [Omega, Omega] | [1, Omega]


Let us use the abbreviations EGF and CGF for extremum
grade form and correlated grade form, respectively. I
use the square brackets to denote that the EGF and CGF
are equivalence classes of ordered pairs. Then you
will notice that the EGF of twisted multivectors
rather directly corresponds to the standard notation
of differential geometry except for the reversal of
the order of the elements in the ordered pairs
(although that notation is usually applied to twisted
forms rather than multivectors). The EGF is a symbolic
analog of one type of icon that Burk uses to represent
twisted multivectors (see, for example, the left hand
side of Figure 28.10 of "Applied Differential
Geometry"), again with a reversal of the two
components of the ordered pair. Now the CGF is a
symbolic analog of the other form of icon on the right
hand side of Burke's Figure 28.10. Very importantly
this icon represents the natural or native geometrical
structure of a twisted multivector as it arises in the
fundamental measurement procedures of physical
theories. For examples from electromagnetism see
Tonti, "On the Geometrical Structure of
Electromagnetism," Figures 1(c) and (d) and
accompanying text (available at
http://discretephysics.dic.units.it/papers/TONTI/Electrom%20Algebraic%20Topology.pdf).

The order chosen for the components of the EGF and CGF
is meant to mimic the familiar placement of the sign
(+ or -) in front of a numerical quantity. The first
component in these forms is then a generalized sign
which represents what Jancewicz has called (following
Lounesto) a "direction" comprising an attitude and
orientation. I have chosen the second component to be
a  carrier of both a direction and relative magnitude.
Thus, these notations are redundunt. Any extended
(straight or twisted) multivector may be represented
as a linear combination of basis multivectors in EGF
or CGF form.

The extremum grade form is so named because its
first component is always either of grade zero or the
largest grade possible in the space under
oonsideration, say, n. The correlated grade form is so
named because its two components' grades are
correlated so that either they are equal or they sum
to n.

The interesting thing is that one can define a
generalized exterior product among straight and
twisted multivectors represented in the CGF. The
second components of the CGF multiply as the normal
exterior product of straight multivectors. However,
one finds that the first components must multiply as
the product of the new algebra -- the OC product. The
OC product is not only essential to the exterior
product of straight and twisted multivectors as
represented in the CGF, but also is key to the
interconversion of the EGF and CGF. 

There are a more complications, but I will end here.