Odd numbered pages should always be right pages!
So you should always use \cleardoublepage before changing page numbering, because it makes no sense to change page numbering on the left side of a double sided print:
\documentclass[twoside]{article}% It's better to tell already the class to use twoside!
\usepackage[showframe]{geometry}
\begin{document}
This is the first page; it should be a right page (recto).
\cleardoublepage
\pagenumbering{arabic}
This is the third page; it should be a right page, because changing page numbering should always be done on right pages!
\end{document}
If you simply wanted to suppress page number at the first page of an article, try:
\documentclass[twoside]{article}% It's better to tell already the class to use twoside!
\usepackage[showframe]{geometry}
\begin{document}
\title{Title}
\author{Me}
\maketitle\thispagestyle{empty}
\begin{abstract}A short abstract of the document.\end{abstract}
Some more text, e.g. a table of contents.
\clearpage
This is the second page; it should be a left page and numbered 2!
\end{document}
Note: You may use \cleardoublepage before \pagenumbering even at oneside documents, because in this case it would have the same result \clearpage would have.
But, if you really, really must have odd pages left and even pages right, you may simply switch the margins (and the margin used for \marginpar):
\documentclass[twoside]{article}% It's better to tell already the class to use twoside!
\usepackage[showframe]{geometry}
\makeatletter
\newcommand*{\flipmargins}{%
\clearpage
\setlength{\@tempdima}{\oddsidemargin}%
\setlength{\oddsidemargin}{\evensidemargin}%
\setlength{\evensidemargin}{\@tempdima}%
\if@reversemargin
\normalmarginpar
\else
\reversemarginpar
\fi
}
\makeatother
\begin{document}
This is the first page; it should be a right page (recto).
\flipmargins
\pagenumbering{roman}
This is the second page; it should be a left page (verso).
\end{document}
Using a page style like headings you may also need to change the page style, e.g., to print the numbers near the outer margin. You may do this, using either scrpage2 or fancyhrd. Here an example using scrpage2:
\documentclass[twoside]{article}% It's better to tell already the class to use twoside!
\usepackage[showframe]{geometry}
\usepackage[automark]{scrpage2}
\pagestyle{scrheadings}
\makeatletter
\newcommand*{\flipmargins}{%
\clearpage
\setlength{\@tempdima}{\oddsidemargin}%
\setlength{\oddsidemargin}{\evensidemargin}%
\setlength{\evensidemargin}{\@tempdima}%
\if@reversemargin
\normalmarginpar
\else
\reversemarginpar
\fi
\ifdim \evensidemargin>\oddsidemargin
\ihead{\headmark}%
\ohead[\pagemark]{\pagemark}%
\else
\ihead[\pagemark]{\pagemark}%
\ohead{\headmark}%
\fi
}
\makeatother
\begin{document}
\section{Normal pages}
This is the first page; it should be a right page (recto).
\flipmargins
\pagenumbering{roman}
\section{Inverse margin pages}
This is the second page; it should be a left page (verso).
\end{document}
Alternative:
One completely different alternative would be to fake the page number, given by \thepage:
\documentclass[twoside]{article}% It's better to tell already the class to use twoside!
\usepackage[showframe]{geometry}
\makeatletter
\newcommand{\@arabicplusone}[1]{\@arabic{\numexpr #1+1\relax}}
\newcommand{\@romanplusone}[1]{\@roman{\numexpr #1+1\relax}}
\newcommand{\@Romanplusone}[1]{\@Roman{\numexpr #1+1\relax}}
\newcommand{\@alphplusone}[1]{\@alph{\numexpr #1+1\relax}}
\newcommand{\@Alphplusone}[1]{\@Alph{\numexpr #1+1\relax}}
\makeatother
\newcommand*{\pagenumberingplusone}[1]{%
\pagenumbering{#1plusone}\setcounter{page}{0}%
}
\begin{document}
This is the first page; it should be a right page (recto). See
page~\pageref{test}.
\clearpage\pagenumberingplusone{roman}
This\label{test} is the second page; it should be a left page (verso).
\end{document}
Disadvantage of this suggestion would be, that if a package uses page counter directly for something else but testing odd or even the result may be unexpected. The advantage of this suggestion would be, that pages with odd value of counter page shows not only an even page number but will be handled to be a even=left page by LaTeX and every class or package that uses \c@page or \value{page} for the test (and vica versa).
Normally, one would simply use the left and right keys, because they are actually synonyms for inner and outer in twoside mode. (Alternatively, one could specify the textwidth and hmarginratio keys.) In case you want to disable the swapping of margins on alternate pages, add the asymmetric option.
\documentclass[twoside]{article}
\usepackage[top=2cm,bottom=4cm,left=1.5cm,right=3cm,asymmetric]{geometry}
\usepackage{lipsum}
\begin{document}
\lipsum[1-15]
\end{document}
EDIT: So far you've used the twoside option in your document class, but you want to do without this option? Well, use the twoside option of geometry instead.
Best Answer
You should use the following setup:
where
<further options>might include12pt.Note that whereas the instructions appear to be phrased as using inches as the main unit of length, your code appears to be using centimeters as the unit of length. Do clarify with your school which length unit applies.